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In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
, an eigenvector () or characteristic vector of a
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, the eigenvector is not rotated.


Formal definition

If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root associated with . There is a direct correspondence between ''n''-by-''n'' square matrices and linear transformations from an ''n''-dimensional vector space into itself, given any basis of the vector space. Hence, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices, or the language of linear transformations. If is finite-dimensional, the above equation is equivalent to A\mathbf = \lambda \mathbf. where is the matrix representation of and is the
coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensio ...
of .


Overview

Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix '' eigen-'' is adopted from the
German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ge ...
word ''
eigen Eigen may refer to: * Eigen (C++ library), computer programming library for matrix and linear algebra operations * Eigen Technologies, the Document AI software company * Eigen, Schwyz, settlement in the municipality of Alpthal in the canton of S ...
'' (
cognate In historical linguistics, cognates or lexical cognates are sets of words in different languages that have been inherited in direct descent from an etymological ancestor in a common parent language. Because language change can have radical ef ...
with the
English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ...
word ''
own Ownership is the state or fact of legal possession and control over property, which may be any asset, tangible or intangible. Ownership can involve multiple rights, collectively referred to as title, which may be separated and held by different ...
'') for 'proper', 'characteristic', 'own'. Originally used to study principal axes of the rotational motion of rigid bodies, eigenvalues and eigenvectors have a wide range of applications, for example in
stability analysis Stability may refer to: Mathematics * Stability theory, the study of the stability of solutions to differential equations and dynamical systems **Asymptotic stability **Linear stability **Lyapunov stability **Orbital stability **Structural stabili ...
,
vibration analysis Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic, such as the motion of a pendulum—or random, such ...
,
atomic orbital In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any ...
s, facial recognition, and
matrix diagonalization In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) ...
. In essence, an eigenvector v of a linear transformation ''T'' is a nonzero vector that, when ''T'' is applied to it, does not change direction. Applying ''T'' to the eigenvector only scales the eigenvector by the scalar value ''λ'', called an eigenvalue. This condition can be written as the equation T(\mathbf) = \lambda \mathbf, referred to as the eigenvalue equation or eigenequation. In general, ''λ'' may be any scalar. For example, ''λ'' may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or complex. The
Mona Lisa The ''Mona Lisa'' ( ; it, Gioconda or ; french: Joconde ) is a half-length portrait painting by Italian artist Leonardo da Vinci. Considered an archetypal masterpiece of the Italian Renaissance, it has been described as "the best kno ...
example pictured here provides a simple illustration. Each point on the painting can be represented as a vector pointing from the center of the painting to that point. The linear transformation in this example is called a shear mapping. Points in the top half are moved to the right, and points in the bottom half are moved to the left, proportional to how far they are from the horizontal axis that goes through the middle of the painting. The vectors pointing to each point in the original image are therefore tilted right or left, and made longer or shorter by the transformation. Points ''along'' the horizontal axis do not move at all when this transformation is applied. Therefore, any vector that points directly to the right or left with no vertical component is an eigenvector of this transformation, because the mapping does not change its direction. Moreover, these eigenvectors all have an eigenvalue equal to one, because the mapping does not change their length either. Linear transformations can take many different forms, mapping vectors in a variety of vector spaces, so the eigenvectors can also take many forms. For example, the linear transformation could be a
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
like \tfrac, in which case the eigenvectors are functions called
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s that are scaled by that differential operator, such as \frace^ = \lambda e^. Alternatively, the linear transformation could take the form of an ''n'' by ''n'' matrix, in which case the eigenvectors are ''n'' by 1 matrices. If the linear transformation is expressed in the form of an ''n'' by ''n'' matrix ''A'', then the eigenvalue equation for a linear transformation above can be rewritten as the matrix multiplication A\mathbf v = \lambda \mathbf v, where the eigenvector ''v'' is an ''n'' by 1 matrix. For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix—for example by
diagonalizing In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
it. Eigenvalues and eigenvectors give rise to many closely related mathematical concepts, and the prefix ''eigen-'' is applied liberally when naming them: * The set of all eigenvectors of a linear transformation, each paired with its corresponding eigenvalue, is called the eigensystem of that transformation. * The set of all eigenvectors of ''T'' corresponding to the same eigenvalue, together with the zero vector, is called an eigenspace, or the characteristic space of ''T'' associated with that eigenvalue. * If a set of eigenvectors of ''T'' forms a basis of the domain of ''T'', then this basis is called an eigenbasis.


History

Eigenvalues are often introduced in the context of
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
or matrix theory. Historically, however, they arose in the study of
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
s and
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s. In the 18th century,
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
studied the rotational motion of a
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
, and discovered the importance of the principal axes.
Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaAugustin-Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. H ...
saw how their work could be used to classify the
quadric surface In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections ( ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is ...
s, and generalized it to arbitrary dimensions. Cauchy also coined the term ''racine caractéristique'' (characteristic root), for what is now called ''eigenvalue''; his term survives in '' characteristic equation''. Later,
Joseph Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and ha ...
used the work of Lagrange and
Pierre-Simon Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarize ...
to solve the
heat equation In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for ...
by
separation of variables In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
in his famous 1822 book '' Théorie analytique de la chaleur''. Charles-François Sturm developed Fourier's ideas further, and brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that real symmetric matrices have real eigenvalues. This was extended by
Charles Hermite Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. ...
in 1855 to what are now called Hermitian matrices. Around the same time, Francesco Brioschi proved that the eigenvalues of orthogonal matrices lie on the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
, and
Alfred Clebsch Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Berlin. ...
found the corresponding result for skew-symmetric matrices. Finally,
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
clarified an important aspect in the
stability theory In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial diffe ...
started by Laplace, by realizing that defective matrices can cause instability. In the meantime, Joseph Liouville studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called ''
Sturm–Liouville theory In mathematics and its applications, classical Sturm–Liouville theory is the theory of ''real'' second-order ''linear'' ordinary differential equations of the form: for given coefficient functions , , and , an unknown function ''y = y''(''x'') ...
''. Schwarz studied the first eigenvalue of
Laplace's equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \na ...
on general domains towards the end of the 19th century, while Poincaré studied
Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with ...
a few years later. At the start of the 20th century,
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
studied the eigenvalues of integral operators by viewing the operators as infinite matrices. He was the first to use the
German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ge ...
word ''eigen'', which means "own", to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by
Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Associat ...
. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is the standard today. The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when
Richard von Mises Richard Edler von Mises (; 19 April 1883 – 14 July 1953) was an Austrian scientist and mathematician who worked on solid mechanics, fluid mechanics, aerodynamics, aeronautics, statistics and probability theory. He held the position of Gordo ...
published the power method. One of the most popular methods today, the QR algorithm, was proposed independently by
John G. F. Francis John G.F. Francis (born 1934) is an English computer scientist, who in 1961 published the QR algorithm for computing the eigenvalues and eigenvectors of matrices, which has been named as one of the ten most important algorithms of the twentieth ...
and Vera Kublanovskaya in 1961.


Eigenvalues and eigenvectors of matrices

Eigenvalues and eigenvectors are often introduced to students in the context of linear algebra courses focused on matrices.Cornell University Department of Mathematics (2016
''Lower-Level Courses for Freshmen and Sophomores''
Accessed on 2016-03-27.
University of Michigan Mathematics (2016
''Math Course Catalogue''
. Accessed on 2016-03-27.
Furthermore, linear transformations over a finite-dimensional vector space can be represented using matrices, which is especially common in numerical and computational applications. Consider -dimensional vectors that are formed as a list of scalars, such as the three-dimensional vectors \mathbf x = \begin1\\-3\\4\end\quad\mbox\quad \mathbf y = \begin-20\\60\\-80\end. These vectors are said to be scalar multiples of each other, or
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster o ...
or
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
, if there is a scalar such that \mathbf x = \lambda \mathbf y. In this case \lambda = -\frac . Now consider the linear transformation of -dimensional vectors defined by an by matrix , A \mathbf v = \mathbf w, or \begin A_ & A_ & \cdots & A_ \\ A_ & A_ & \cdots & A_ \\ \vdots & \vdots & \ddots & \vdots \\ A_ & A_ & \cdots & A_ \\ \end\begin v_1 \\ v_2 \\ \vdots \\ v_n \end = \begin w_1 \\ w_2 \\ \vdots \\ w_n \end where, for each row, w_i = A_ v_1 + A_ v_2 + \cdots + A_ v_n = \sum_^n A_ v_j. If it occurs that and are scalar multiples, that is if then is an eigenvector of the linear transformation and the scale factor is the eigenvalue corresponding to that eigenvector. Equation () is the eigenvalue equation for the matrix . Equation () can be stated equivalently as where is the by
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
and 0 is the zero vector.


Eigenvalues and the characteristic polynomial

Equation () has a nonzero solution ''v''
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the matrix is zero. Therefore, the eigenvalues of ''A'' are values of ''λ'' that satisfy the equation Using the Leibniz formula for determinants, the left-hand side of Equation () is a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
function of the variable ''λ'' and the degree of this polynomial is ''n'', the order of the matrix ''A''. Its
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s depend on the entries of ''A'', except that its term of degree ''n'' is always (−1)''n''''λ''''n''. This polynomial is called the ''
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The c ...
'' of ''A''. Equation () is called the ''characteristic equation'' or the ''secular equation'' of ''A''. The
fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
implies that the characteristic polynomial of an ''n''-by-''n'' matrix ''A'', being a polynomial of degree ''n'', can be factored into the product of ''n'' linear terms, where each ''λ''''i'' may be real but in general is a complex number. The numbers ''λ''1, ''λ''2, ..., ''λ''''n'', which may not all have distinct values, are roots of the polynomial and are the eigenvalues of ''A''. As a brief example, which is described in more detail in the examples section later, consider the matrix A = \begin 2 & 1\\ 1 & 2 \end. Taking the determinant of , the characteristic polynomial of ''A'' is , A - \lambda I, = \begin 2 - \lambda & 1 \\ 1 & 2 - \lambda \end = 3 - 4\lambda + \lambda^2. Setting the characteristic polynomial equal to zero, it has roots at and , which are the two eigenvalues of ''A''. The eigenvectors corresponding to each eigenvalue can be found by solving for the components of v in the equation In this example, the eigenvectors are any nonzero scalar multiples of \mathbf v_ = \begin 1 \\ -1 \end, \quad \mathbf v_ = \begin 1 \\ 1 \end. If the entries of the matrix ''A'' are all real numbers, then the coefficients of the characteristic polynomial will also be real numbers, but the eigenvalues may still have nonzero imaginary parts. The entries of the corresponding eigenvectors therefore may also have nonzero imaginary parts. Similarly, the eigenvalues may be
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
s even if all the entries of ''A'' are
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s or even if they are all integers. However, if the entries of ''A'' are all
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of th ...
s, which include the rationals, the eigenvalues are complex algebraic numbers. The non-real roots of a real polynomial with real coefficients can be grouped into pairs of
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
s, namely with the two members of each pair having imaginary parts that differ only in sign and the same real part. If the degree is odd, then by the
intermediate value theorem In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two impor ...
at least one of the roots is real. Therefore, any real matrix with odd order has at least one real eigenvalue, whereas a real matrix with even order may not have any real eigenvalues. The eigenvectors associated with these complex eigenvalues are also complex and also appear in complex conjugate pairs.


Algebraic multiplicity

Let ''λ''''i'' be an eigenvalue of an ''n'' by ''n'' matrix ''A''. The algebraic multiplicity ''μ''''A''(''λ''''i'') of the eigenvalue is its multiplicity as a root of the characteristic polynomial, that is, the largest integer ''k'' such that (''λ'' − ''λ''''i'')''k'' divides evenly that polynomial. Suppose a matrix ''A'' has dimension ''n'' and ''d'' ≤ ''n'' distinct eigenvalues. Whereas Equation () factors the characteristic polynomial of ''A'' into the product of ''n'' linear terms with some terms potentially repeating, the characteristic polynomial can instead be written as the product of ''d'' terms each corresponding to a distinct eigenvalue and raised to the power of the algebraic multiplicity, , A - \lambda I, = (\lambda_1 - \lambda)^(\lambda_2 - \lambda)^ \cdots (\lambda_d - \lambda)^. If ''d'' = ''n'' then the right-hand side is the product of ''n'' linear terms and this is the same as Equation (). The size of each eigenvalue's algebraic multiplicity is related to the dimension ''n'' as \begin 1 &\leq \mu_A(\lambda_i) \leq n, \\ \mu_A &= \sum_^d \mu_A\left(\lambda_i\right) = n. \end If ''μ''''A''(''λ''''i'') = 1, then ''λ''''i'' is said to be a ''simple eigenvalue''. If ''μ''''A''(''λ''''i'') equals the geometric multiplicity of ''λ''''i'', ''γ''''A''(''λ''''i''), defined in the next section, then ''λ''''i'' is said to be a ''semisimple eigenvalue''.


Eigenspaces, geometric multiplicity, and the eigenbasis for matrices

Given a particular eigenvalue ''λ'' of the ''n'' by ''n'' matrix ''A'', define the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
''E'' to be all vectors v that satisfy Equation (), E = \left\. On one hand, this set is precisely the kernel or nullspace of the matrix (''A'' − ''λI''). On the other hand, by definition, any nonzero vector that satisfies this condition is an eigenvector of ''A'' associated with ''λ''. So, the set ''E'' is the union of the zero vector with the set of all eigenvectors of ''A'' associated with ''λ'', and ''E'' equals the nullspace of (''A'' − ''λI''). ''E'' is called the eigenspace or characteristic space of ''A'' associated with ''λ''. In general ''λ'' is a complex number and the eigenvectors are complex ''n'' by 1 matrices. A property of the nullspace is that it is a
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
, so ''E'' is a linear subspace of \mathbb^n. Because the eigenspace ''E'' is a linear subspace, it is
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
under addition. That is, if two vectors u and v belong to the set ''E'', written , then or equivalently . This can be checked using the
distributive property In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmet ...
of matrix multiplication. Similarly, because ''E'' is a linear subspace, it is closed under scalar multiplication. That is, if and ''α'' is a complex number, or equivalently . This can be checked by noting that multiplication of complex matrices by complex numbers is
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
. As long as u + v and ''α''v are not zero, they are also eigenvectors of ''A'' associated with ''λ''. The dimension of the eigenspace ''E'' associated with ''λ'', or equivalently the maximum number of linearly independent eigenvectors associated with ''λ'', is referred to as the eigenvalue's geometric multiplicity ''γ''''A''(''λ''). Because ''E'' is also the nullspace of (''A'' − ''λI''), the geometric multiplicity of ''λ'' is the dimension of the nullspace of (''A'' − ''λI''), also called the ''nullity'' of (''A'' − ''λI''), which relates to the dimension and rank of (''A'' − ''λI'') as \gamma_A(\lambda) = n - \operatorname(A - \lambda I). Because of the definition of eigenvalues and eigenvectors, an eigenvalue's geometric multiplicity must be at least one, that is, each eigenvalue has at least one associated eigenvector. Furthermore, an eigenvalue's geometric multiplicity cannot exceed its algebraic multiplicity. Additionally, recall that an eigenvalue's algebraic multiplicity cannot exceed ''n''. 1 \le \gamma_A(\lambda) \le \mu_A(\lambda) \le n To prove the inequality \gamma_A(\lambda)\le\mu_A(\lambda), consider how the definition of geometric multiplicity implies the existence of \gamma_A(\lambda)
orthonormal In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of ...
eigenvectors \boldsymbol_1,\, \ldots,\, \boldsymbol_, such that A \boldsymbol_k = \lambda \boldsymbol_k. We can therefore find a (unitary) matrix V whose first \gamma_A(\lambda) columns are these eigenvectors, and whose remaining columns can be any orthonormal set of n - \gamma_A(\lambda) vectors orthogonal to these eigenvectors of A. Then V has full rank and is therefore invertible, and AV=VD with D a matrix whose top left block is the diagonal matrix \lambda I_. This implies that (A - \xi I)V = V(D - \xi I). In other words, A - \xi I is similar to D - \xi I, which implies that \det(A - \xi I) = \det(D - \xi I). But from the definition of D we know that \det(D - \xi I) contains a factor (\xi - \lambda)^, which means that the algebraic multiplicity of \lambda must satisfy \mu_A(\lambda) \ge \gamma_A(\lambda). Suppose A has d \leq n distinct eigenvalues \lambda_1, \ldots, \lambda_d, where the geometric multiplicity of \lambda_i is \gamma_A (\lambda_i). The total geometric multiplicity of A, \begin \gamma_A &= \sum_^d \gamma_A(\lambda_i), \\ d &\le \gamma_A \le n, \end is the dimension of the sum of all the eigenspaces of A's eigenvalues, or equivalently the maximum number of linearly independent eigenvectors of A. If \gamma_A=n, then * The direct sum of the eigenspaces of all of A's eigenvalues is the entire vector space \mathbb^n. * A basis of \mathbb^n can be formed from n linearly independent eigenvectors of A; such a basis is called an eigenbasis * Any vector in \mathbb^n can be written as a linear combination of eigenvectors of A.


Additional properties of eigenvalues

Let A be an arbitrary n \times n matrix of complex numbers with eigenvalues \lambda_1, \ldots, \lambda_n. Each eigenvalue appears \mu_A(\lambda_i) times in this list, where \mu_A(\lambda_i) is the eigenvalue's algebraic multiplicity. The following are properties of this matrix and its eigenvalues: * The trace of A, defined as the sum of its diagonal elements, is also the sum of all eigenvalues, *: \operatorname(A) = \sum_^n a_ = \sum_^n \lambda_i = \lambda_1 + \lambda_2 + \cdots + \lambda_n. * The
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of A is the product of all its eigenvalues, *: \det(A) = \prod_^n \lambda_i = \lambda_1\lambda_2 \cdots \lambda_n. * The eigenvalues of the kth power of A; i.e., the eigenvalues of A^k, for any positive integer k, are \lambda_1^k, \ldots, \lambda_n^k. * The matrix A is invertible if and only if every eigenvalue is nonzero. * If A is invertible, then the eigenvalues of A^ are \frac, \ldots, \frac and each eigenvalue's geometric multiplicity coincides. Moreover, since the characteristic polynomial of the inverse is the reciprocal polynomial of the original, the eigenvalues share the same algebraic multiplicity. * If A is equal to its
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
A^*, or equivalently if A is
Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
, then every eigenvalue is real. The same is true of any symmetric real matrix. * If A is not only Hermitian but also positive-definite, positive-semidefinite, negative-definite, or negative-semidefinite, then every eigenvalue is positive, non-negative, negative, or non-positive, respectively. * If A is
unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation In mathematics, a unitary representation of a grou ...
, every eigenvalue has absolute value , \lambda_i, =1. * if A is a n\times n matrix and \ are its eigenvalues, then the eigenvalues of matrix I+A (where I is the identity matrix) are \. Moreover, if \alpha\in\mathbb C, the eigenvalues of \alpha I+A are \. More generally, for a polynomial P the eigenvalues of matrix P(A) are \.


Left and right eigenvectors

Many disciplines traditionally represent vectors as matrices with a single column rather than as matrices with a single row. For that reason, the word "eigenvector" in the context of matrices almost always refers to a right eigenvector, namely a ''column'' vector that ''right'' multiplies the n \times n matrix A in the defining equation, Equation (), A \mathbf v = \lambda \mathbf v. The eigenvalue and eigenvector problem can also be defined for ''row'' vectors that ''left'' multiply matrix A. In this formulation, the defining equation is \mathbf u A = \kappa \mathbf u, where \kappa is a scalar and u is a 1 \times n matrix. Any row vector u satisfying this equation is called a left eigenvector of A and \kappa is its associated eigenvalue. Taking the transpose of this equation, A^\textsf \mathbf u^\textsf = \kappa \mathbf u^\textsf. Comparing this equation to Equation (), it follows immediately that a left eigenvector of A is the same as the transpose of a right eigenvector of A^\textsf, with the same eigenvalue. Furthermore, since the characteristic polynomial of A^\textsf is the same as the characteristic polynomial of A, the eigenvalues of the left eigenvectors of A are the same as the eigenvalues of the right eigenvectors of A^\textsf.


Diagonalization and the eigendecomposition

Suppose the eigenvectors of ''A'' form a basis, or equivalently ''A'' has ''n'' linearly independent eigenvectors v1, v2, ..., v''n'' with associated eigenvalues ''λ''1, ''λ''2, ..., ''λ''''n''. The eigenvalues need not be distinct. Define a
square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...
''Q'' whose columns are the ''n'' linearly independent eigenvectors of ''A'', : Q = \begin \mathbf v_1 & \mathbf v_2 & \cdots & \mathbf v_n \end. Since each column of ''Q'' is an eigenvector of ''A'', right multiplying ''A'' by ''Q'' scales each column of ''Q'' by its associated eigenvalue, : AQ = \begin \lambda_1 \mathbf v_1 & \lambda_2 \mathbf v_2 & \cdots & \lambda_n \mathbf v_n \end. With this in mind, define a diagonal matrix Λ where each diagonal element Λ''ii'' is the eigenvalue associated with the ''i''th column of ''Q''. Then : AQ = Q\Lambda. Because the columns of ''Q'' are linearly independent, Q is invertible. Right multiplying both sides of the equation by ''Q''−1, : A = Q\Lambda Q^, or by instead left multiplying both sides by ''Q''−1, : Q^AQ = \Lambda. ''A'' can therefore be decomposed into a matrix composed of its eigenvectors, a diagonal matrix with its eigenvalues along the diagonal, and the inverse of the matrix of eigenvectors. This is called the eigendecomposition and it is a similarity transformation. Such a matrix ''A'' is said to be ''similar'' to the diagonal matrix Λ or '' diagonalizable''. The matrix ''Q'' is the change of basis matrix of the similarity transformation. Essentially, the matrices ''A'' and Λ represent the same linear transformation expressed in two different bases. The eigenvectors are used as the basis when representing the linear transformation as Λ. Conversely, suppose a matrix ''A'' is diagonalizable. Let ''P'' be a non-singular square matrix such that ''P''−1''AP'' is some diagonal matrix ''D''. Left multiplying both by ''P'', . Each column of ''P'' must therefore be an eigenvector of ''A'' whose eigenvalue is the corresponding diagonal element of ''D''. Since the columns of ''P'' must be linearly independent for ''P'' to be invertible, there exist ''n'' linearly independent eigenvectors of ''A''. It then follows that the eigenvectors of ''A'' form a basis if and only if ''A'' is diagonalizable. A matrix that is not diagonalizable is said to be defective. For defective matrices, the notion of eigenvectors generalizes to generalized eigenvectors and the diagonal matrix of eigenvalues generalizes to the Jordan normal form. Over an algebraically closed field, any matrix ''A'' has a Jordan normal form and therefore admits a basis of generalized eigenvectors and a decomposition into generalized eigenspaces.


Variational characterization

In the
Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
case, eigenvalues can be given a variational characterization. The largest eigenvalue of H is the maximum value of the
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
\mathbf x^\textsf H \mathbf x / \mathbf x^\textsf \mathbf x. A value of \mathbf x that realizes that maximum, is an eigenvector.


Matrix examples


Two-dimensional matrix example

Consider the matrix A = \begin 2 & 1\\ 1 & 2 \end. The figure on the right shows the effect of this transformation on point coordinates in the plane. The eigenvectors ''v'' of this transformation satisfy Equation (), and the values of ''λ'' for which the determinant of the matrix (''A'' − ''λI'') equals zero are the eigenvalues. Taking the determinant to find characteristic polynomial of ''A'', \begin , A - \lambda I, &= \left, \begin 2 & 1 \\ 1 & 2 \end - \lambda\begin 1 & 0 \\ 0 & 1 \end\ = \begin 2 - \lambda & 1 \\ 1 & 2 - \lambda \end \\ pt &= 3 - 4\lambda + \lambda^2 \\ pt &= (\lambda - 3)(\lambda - 1). \end Setting the characteristic polynomial equal to zero, it has roots at and , which are the two eigenvalues of ''A''. For , Equation () becomes, (A - I)\mathbf_ = \begin 1 & 1\\ 1 & 1\end\beginv_1 \\ v_2\end = \begin0 \\ 0\end 1v_1 + 1v_2 = 0 Any nonzero vector with ''v''1 = −''v''2 solves this equation. Therefore, \mathbf_ = \begin v_1 \\ -v_1 \end = \begin 1 \\ -1 \end is an eigenvector of ''A'' corresponding to ''λ'' = 1, as is any scalar multiple of this vector. For , Equation () becomes \begin (A - 3I)\mathbf_ &= \begin -1 & 1\\ 1 & -1 \end \begin v_1 \\ v_2 \end = \begin 0 \\ 0 \end \\ -1v_1 + 1v_2 &= 0;\\ 1v_1 - 1v_2 &= 0 \end Any nonzero vector with ''v''1 = ''v''2 solves this equation. Therefore, \mathbf v_ = \begin v_1 \\ v_1 \end = \begin 1 \\ 1 \end is an eigenvector of ''A'' corresponding to ''λ'' = 3, as is any scalar multiple of this vector. Thus, the vectors v''λ''=1 and v''λ''=3 are eigenvectors of ''A'' associated with the eigenvalues and , respectively.


Three-dimensional matrix example

Consider the matrix A = \begin 2 & 0 & 0 \\ 0 & 3 & 4 \\ 0 & 4 & 9 \end. The characteristic polynomial of ''A'' is \begin , A-\lambda I, &= \left, \begin 2 & 0 & 0 \\ 0 & 3 & 4 \\ 0 & 4 & 9 \end - \lambda\begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end\ = \begin 2 - \lambda & 0 & 0 \\ 0 & 3 - \lambda & 4 \\ 0 & 4 & 9 - \lambda \end, \\ pt &= (2 - \lambda)\bigl 3 - \lambda)(9 - \lambda) - 16\bigr = -\lambda^3 + 14\lambda^2 - 35\lambda + 22. \end The roots of the characteristic polynomial are 2, 1, and 11, which are the only three eigenvalues of ''A''. These eigenvalues correspond to the eigenvectors and or any nonzero multiple thereof.


Three-dimensional matrix example with complex eigenvalues

Consider the cyclic permutation matrix A = \begin 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0 \end. This matrix shifts the coordinates of the vector up by one position and moves the first coordinate to the bottom. Its characteristic polynomial is 1 − ''λ''3, whose roots are \begin \lambda_1 &= 1 \\ \lambda_2 &= -\frac + i \frac \\ \lambda_3 &= \lambda_2^* = -\frac - i \frac \end where i is an
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
with For the real eigenvalue ''λ''1 = 1, any vector with three equal nonzero entries is an eigenvector. For example, A \begin 5\\ 5\\ 5 \end = \begin 5\\ 5\\ 5 \end = 1 \cdot \begin 5\\ 5\\ 5 \end. For the complex conjugate pair of imaginary eigenvalues, \lambda_2\lambda_3 = 1, \quad \lambda_2^2 = \lambda_3, \quad \lambda_3^2 = \lambda_2. Then A \begin 1 \\ \lambda_2 \\ \lambda_3 \end = \begin \lambda_2 \\ \lambda_3 \\ 1 \end = \lambda_2 \cdot \begin 1 \\ \lambda_2 \\ \lambda_3 \end, and A \begin 1 \\ \lambda_3 \\ \lambda_2 \end = \begin \lambda_3 \\ \lambda_2 \\ 1 \end = \lambda_3 \cdot \begin 1 \\ \lambda_3 \\ \lambda_2 \end. Therefore, the other two eigenvectors of ''A'' are complex and are \mathbf v_ = \begin 1 & \lambda_2 & \lambda_3\end^\textsf and \mathbf v_ = \begin 1 & \lambda_3 & \lambda_2\end^\textsf with eigenvalues ''λ''2 and ''λ''3, respectively. The two complex eigenvectors also appear in a complex conjugate pair, \mathbf v_ = \mathbf v_^*.


Diagonal matrix example

Matrices with entries only along the main diagonal are called '' diagonal matrices''. The eigenvalues of a diagonal matrix are the diagonal elements themselves. Consider the matrix A = \begin 1 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 3\end. The characteristic polynomial of ''A'' is , A - \lambda I, = (1 - \lambda)(2 - \lambda)(3 - \lambda), which has the roots , , and . These roots are the diagonal elements as well as the eigenvalues of ''A''. Each diagonal element corresponds to an eigenvector whose only nonzero component is in the same row as that diagonal element. In the example, the eigenvalues correspond to the eigenvectors, \mathbf v_ = \begin 1\\ 0\\ 0 \end,\quad \mathbf v_ = \begin 0\\ 1\\ 0 \end,\quad \mathbf v_ = \begin 0\\ 0\\ 1 \end, respectively, as well as scalar multiples of these vectors.


Triangular matrix example

A matrix whose elements above the main diagonal are all zero is called a ''lower triangular matrix'', while a matrix whose elements below the main diagonal are all zero is called an ''upper triangular matrix''. As with diagonal matrices, the eigenvalues of triangular matrices are the elements of the main diagonal. Consider the lower triangular matrix, A = \begin 1 & 0 & 0\\ 1 & 2 & 0\\ 2 & 3 & 3 \end. The characteristic polynomial of ''A'' is , A - \lambda I, = (1 - \lambda)(2 - \lambda)(3 - \lambda), which has the roots , , and . These roots are the diagonal elements as well as the eigenvalues of ''A''. These eigenvalues correspond to the eigenvectors, \mathbf v_ = \begin 1\\ -1\\ \frac\end,\quad \mathbf v_ = \begin 0\\ 1\\ -3\end,\quad \mathbf v_ = \begin 0\\ 0\\ 1\end, respectively, as well as scalar multiples of these vectors.


Matrix with repeated eigenvalues example

As in the previous example, the lower triangular matrix A = \begin 2 & 0 & 0 & 0 \\ 1 & 2 & 0 & 0 \\ 0 & 1 & 3 & 0 \\ 0 & 0 & 1 & 3 \end, has a characteristic polynomial that is the product of its diagonal elements, , A - \lambda I, = \begin 2 - \lambda & 0 & 0 & 0 \\ 1 & 2- \lambda & 0 & 0 \\ 0 & 1 & 3- \lambda & 0 \\ 0 & 0 & 1 & 3- \lambda \end = (2 - \lambda)^2(3 - \lambda)^2. The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The ''algebraic multiplicity'' of each eigenvalue is 2; in other words they are both double roots. The sum of the algebraic multiplicities of all distinct eigenvalues is ''μ''''A'' = 4 = ''n'', the order of the characteristic polynomial and the dimension of ''A''. On the other hand, the ''geometric multiplicity'' of the eigenvalue 2 is only 1, because its eigenspace is spanned by just one vector \begin 0 & 1 & -1 & 1 \end^\textsf and is therefore 1-dimensional. Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by just one vector \begin 0 & 0 & 0 & 1 \end^\textsf. The total geometric multiplicity ''γ''''A'' is 2, which is the smallest it could be for a matrix with two distinct eigenvalues. Geometric multiplicities are defined in a later section.


Eigenvector-eigenvalue identity

For a Hermitian matrix, the norm squared of the ''j''th component of a normalized eigenvector can be calculated using only the matrix eigenvalues and the eigenvalues of the corresponding minor matrix, , v_, ^2 = \frac, where M_j is the submatrix formed by removing the ''j''th row and column from the original matrix. This identity also extends to diagonalizable matrices, and has been rediscovered many times in the literature.


Eigenvalues and eigenfunctions of differential operators

The definitions of eigenvalue and eigenvectors of a linear transformation ''T'' remains valid even if the underlying vector space is an infinite-dimensional Hilbert or
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
. A widely used class of linear transformations acting on infinite-dimensional spaces are the
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
s on
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
s. Let ''D'' be a linear differential operator on the space C of infinitely differentiable real functions of a real argument ''t''. The eigenvalue equation for ''D'' is the
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
D f(t) = \lambda f(t) The functions that satisfy this equation are eigenvectors of ''D'' and are commonly called eigenfunctions.


Derivative operator example

Consider the derivative operator \tfrac with eigenvalue equation \fracf(t) = \lambda f(t). This differential equation can be solved by multiplying both sides by ''dt''/''f''(''t'') and integrating. Its solution, the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
f(t) = f(0)e^, is the eigenfunction of the derivative operator. In this case the eigenfunction is itself a function of its associated eigenvalue. In particular, for ''λ'' = 0 the eigenfunction ''f''(''t'') is a constant. The main
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
article gives other examples.


General definition

The concept of eigenvalues and eigenvectors extends naturally to arbitrary linear transformations on arbitrary vector spaces. Let ''V'' be any vector space over some field ''K'' of
scalars Scalar may refer to: * Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
, and let ''T'' be a linear transformation mapping ''V'' into ''V'', T:V \to V. We say that a nonzero vector v ∈ ''V'' is an eigenvector of ''T'' if and only if there exists a scalar ''λ'' ∈ ''K'' such that This equation is called the eigenvalue equation for ''T'', and the scalar ''λ'' is the eigenvalue of ''T'' corresponding to the eigenvector v. ''T''(v) is the result of applying the transformation ''T'' to the vector v, while ''λ''v is the product of the scalar ''λ'' with v.


Eigenspaces, geometric multiplicity, and the eigenbasis

Given an eigenvalue ''λ'', consider the set E = \left\, which is the union of the zero vector with the set of all eigenvectors associated with ''λ''. ''E'' is called the eigenspace or characteristic space of ''T'' associated with ''λ''. By definition of a linear transformation, \begin T(\mathbf + \mathbf) &= T(\mathbf) + T(\mathbf),\\ T(\alpha \mathbf) &= \alpha T(\mathbf), \end for x, y ∈ ''V'' and ''α'' ∈ ''K''. Therefore, if u and v are eigenvectors of ''T'' associated with eigenvalue ''λ'', namely u, v ∈ ''E'', then \begin T(\mathbf + \mathbf) &= \lambda (\mathbf + \mathbf),\\ T(\alpha \mathbf) &= \lambda (\alpha \mathbf). \end So, both u + v and αv are either zero or eigenvectors of ''T'' associated with ''λ'', namely u + v, ''α''v ∈ ''E'', and ''E'' is closed under addition and scalar multiplication. The eigenspace ''E'' associated with ''λ'' is therefore a linear subspace of ''V''. If that subspace has dimension 1, it is sometimes called an eigenline. The geometric multiplicity ''γ''''T''(''λ'') of an eigenvalue ''λ'' is the dimension of the eigenspace associated with ''λ'', i.e., the maximum number of linearly independent eigenvectors associated with that eigenvalue. By the definition of eigenvalues and eigenvectors, ''γ''''T''(''λ'') ≥ 1 because every eigenvalue has at least one eigenvector. The eigenspaces of ''T'' always form a
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
. As a consequence, eigenvectors of ''different'' eigenvalues are always linearly independent. Therefore, the sum of the dimensions of the eigenspaces cannot exceed the dimension ''n'' of the vector space on which ''T'' operates, and there cannot be more than ''n'' distinct eigenvalues. Any subspace spanned by eigenvectors of ''T'' is an
invariant subspace In mathematics, an invariant subspace of a linear mapping ''T'' : ''V'' → ''V '' i.e. from some vector space ''V'' to itself, is a subspace ''W'' of ''V'' that is preserved by ''T''; that is, ''T''(''W'') ⊆ ''W''. General desc ...
of ''T'', and the restriction of ''T'' to such a subspace is diagonalizable. Moreover, if the entire vector space ''V'' can be spanned by the eigenvectors of ''T'', or equivalently if the direct sum of the eigenspaces associated with all the eigenvalues of ''T'' is the entire vector space ''V'', then a basis of ''V'' called an eigenbasis can be formed from linearly independent eigenvectors of ''T''. When ''T'' admits an eigenbasis, ''T'' is diagonalizable.


Spectral theory

If ''λ'' is an eigenvalue of ''T'', then the operator (''T'' − ''λI'') is not one-to-one, and therefore its inverse (''T'' − ''λI'')−1 does not exist. The converse is true for finite-dimensional vector spaces, but not for infinite-dimensional vector spaces. In general, the operator (''T'' − ''λI'') may not have an inverse even if ''λ'' is not an eigenvalue. For this reason, in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
eigenvalues can be generalized to the spectrum of a linear operator ''T'' as the set of all scalars ''λ'' for which the operator (''T'' − ''λI'') has no bounded inverse. The spectrum of an operator always contains all its eigenvalues but is not limited to them.


Associative algebras and representation theory

One can generalize the algebraic object that is acting on the vector space, replacing a single operator acting on a vector space with an
algebra representation In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint func ...
– an
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
acting on a
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...
. The study of such actions is the field of
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
. The representation-theoretical concept of weight is an analog of eigenvalues, while ''weight vectors'' and ''weight spaces'' are the analogs of eigenvectors and eigenspaces, respectively.


Dynamic equations

The simplest
difference equation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
s have the form : x_t = a_1 x_ + a_2 x_ + \cdots + a_k x_. The solution of this equation for ''x'' in terms of ''t'' is found by using its characteristic equation : \lambda^k - a_1\lambda^ - a_2\lambda^ - \cdots - a_\lambda-a_k = 0, which can be found by stacking into matrix form a set of equations consisting of the above difference equation and the ''k'' – 1 equations x_ = x_,\ \dots,\ x_ = x_, giving a ''k''-dimensional system of the first order in the stacked variable vector \begin x_t & \cdots & x_ \end in terms of its once-lagged value, and taking the characteristic equation of this system's matrix. This equation gives ''k'' characteristic roots \lambda_1,\, \ldots,\, \lambda_k, for use in the solution equation : x_t = c_1\lambda_1^t + \cdots + c_k\lambda_k^t. A similar procedure is used for solving a
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
of the form : \frac + a_\frac + \cdots + a_1\frac + a_0 x = 0.


Calculation

The calculation of eigenvalues and eigenvectors is a topic where theory, as presented in elementary linear algebra textbooks, is often very far from practice.


Classical method

The classical method is to first find the eigenvalues, and then calculate the eigenvectors for each eigenvalue. It is in several ways poorly suited for non-exact arithmetics such as
floating-point In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can ...
.


Eigenvalues

The eigenvalues of a matrix A can be determined by finding the roots of the characteristic polynomial. This is easy for 2 \times 2 matrices, but the difficulty increases rapidly with the size of the matrix. In theory, the coefficients of the characteristic polynomial can be computed exactly, since they are sums of products of matrix elements; and there are algorithms that can find all the roots of a polynomial of arbitrary degree to any required
accuracy Accuracy and precision are two measures of '' observational error''. ''Accuracy'' is how close a given set of measurements ( observations or readings) are to their '' true value'', while ''precision'' is how close the measurements are to each o ...
. However, this approach is not viable in practice because the coefficients would be contaminated by unavoidable
round-off error A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are d ...
s, and the roots of a polynomial can be an extremely sensitive function of the coefficients (as exemplified by
Wilkinson's polynomial In numerical analysis, Wilkinson's polynomial is a specific polynomial which was used by James H. Wilkinson in 1963 to illustrate a difficulty when finding the root of a polynomial: the location of the roots can be very sensitive to perturbatio ...
). Even for matrices whose elements are integers the calculation becomes nontrivial, because the sums are very long; the constant term is the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
, which for an n \times n matrix is a sum of n! different products. Explicit algebraic formulas for the roots of a polynomial exist only if the degree n is 4 or less. According to the Abel–Ruffini theorem there is no general, explicit and exact algebraic formula for the roots of a polynomial with degree 5 or more. (Generality matters because any polynomial with degree n is the characteristic polynomial of some
companion matrix In linear algebra, the Frobenius companion matrix of the monic polynomial : p(t)=c_0 + c_1 t + \cdots + c_t^ + t^n ~, is the square matrix defined as :C(p)=\begin 0 & 0 & \dots & 0 & -c_0 \\ 1 & 0 & \dots & 0 & -c_1 \\ 0 & 1 & \dots & 0 & -c_2 ...
of order n.) Therefore, for matrices of order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula, and must therefore be computed by approximate
numerical method In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. Mathem ...
s. Even the exact formula for the roots of a degree 3 polynomial is numerically impractical.


Eigenvectors

Once the (exact) value of an eigenvalue is known, the corresponding eigenvectors can be found by finding nonzero solutions of the eigenvalue equation, that becomes a
system of linear equations In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in t ...
with known coefficients. For example, once it is known that 6 is an eigenvalue of the matrix A = \begin 4 & 1\\ 6 & 3\end we can find its eigenvectors by solving the equation A v = 6 v, that is \begin 4 & 1\\ 6 & 3\end\beginx \\y\end = 6 \cdot \beginx \\y\end This matrix equation is equivalent to two
linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...
s \left\{ \begin{aligned} 4x + y &= 6x \\ 6x + 3y &= 6y\end{aligned} \right. that is \left\{ \begin{aligned} -2x + y &= 0 \\ 6x - 3y &= 0\end{aligned} \right. Both equations reduce to the single linear equation y=2x. Therefore, any vector of the form \begin{bmatrix} a & 2a \end{bmatrix}^\textsf{T}, for any nonzero real number a, is an eigenvector of A with eigenvalue \lambda = 6. The matrix A above has another eigenvalue \lambda=1. A similar calculation shows that the corresponding eigenvectors are the nonzero solutions of 3x+y=0, that is, any vector of the form \begin{bmatrix} b & -3b \end{bmatrix}^\textsf{T}, for any nonzero real number b.


Simple iterative methods

The converse approach, of first seeking the eigenvectors and then determining each eigenvalue from its eigenvector, turns out to be far more tractable for computers. The easiest algorithm here consists of picking an arbitrary starting vector and then repeatedly multiplying it with the matrix (optionally normalizing the vector to keep its elements of reasonable size); this makes the vector converge towards an eigenvector. A variation is to instead multiply the vector by this causes it to converge to an eigenvector of the eigenvalue closest to If \mathbf{v} is (a good approximation of) an eigenvector of A, then the corresponding eigenvalue can be computed as : \lambda = \frac{\mathbf{v}^* A\mathbf{v{\mathbf{v}^* \mathbf{v where \mathbf{v}^* denotes the
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
of \mathbf{v}.


Modern methods

Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the QR algorithm was designed in 1961. Combining the Householder transformation with the LU decomposition results in an algorithm with better convergence than the QR algorithm. For large
Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
sparse matrices, the
Lanczos algorithm The Lanczos algorithm is an iterative method devised by Cornelius Lanczos that is an adaptation of power methods to find the m "most useful" (tending towards extreme highest/lowest) eigenvalues and eigenvectors of an n \times n Hermitian matr ...
is one example of an efficient
iterative method In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the pre ...
to compute eigenvalues and eigenvectors, among several other possibilities. Most numeric methods that compute the eigenvalues of a matrix also determine a set of corresponding eigenvectors as a by-product of the computation, although sometimes implementors choose to discard the eigenvector information as soon as it is no longer needed.


Applications


Eigenvalues of geometric transformations

The following table presents some example transformations in the plane along with their 2×2 matrices, eigenvalues, and eigenvectors. {, class="wikitable" style="text-align:center; margin:1em auto 1em auto;" , + Eigenvalues of geometric transformations , - ! ! scope="col" , Scaling ! scope="col" , Unequal scaling ! scope="col" ,
Rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
! scope="col" , Horizontal shear ! scope="col" ,
Hyperbolic rotation In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation or shear mapping. For a fixed positive real number , t ...
, - ! scope="row" , Illustration , , , , , , - style="vertical-align:top" ! scope="row" , Matrix , \begin{bmatrix}k & 0\\ 0 & k\end{bmatrix} , \begin{bmatrix}k_1 & 0\\ 0 & k_2\end{bmatrix} , \begin{bmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{bmatrix} , \begin{bmatrix}1 & k\\ 0 & 1\end{bmatrix} , \begin{bmatrix}\cosh\varphi & \sinh\varphi\\ \sinh\varphi & \cosh\varphi\end{bmatrix} , - ! scope="row" , Characteristic
polynomial , \ (\lambda - k)^2 , (\lambda - k_1)(\lambda - k_2) , \lambda^2 - 2\cos(\theta)\lambda + 1 , \ (\lambda - 1)^2 , \lambda^2 - 2\cosh(\varphi)\lambda + 1 , - ! scope="row" , Eigenvalues, \lambda_i , \lambda_1 = \lambda_2 = k , \begin{align}\lambda_1 &= k_1 \\ \lambda_2 &= k_2\end{align} , \begin{align}\lambda_1 &= e^{i\theta} \\ &= \cos\theta + i\sin\theta \\ \lambda_2 &= e^{-i\theta} \\ &= \cos\theta - i\sin\theta \end{align} , \lambda_1 = \lambda_2 = 1 , \begin{align}\lambda_1 &= e^\varphi \\ &= \cosh\varphi + \sinh\varphi \\ \lambda_2 &= e^{-\varphi} \\ &= \cosh\varphi - \sinh\varphi \end{align} , - ! scope="row" , Algebraic ,
\mu_i = \mu(\lambda_i) , \mu_1 = 2 , \begin{align}\mu_1 &= 1 \\ \mu_2 &= 1 \end{align} , \begin{align}\mu_1 &= 1 \\ \mu_2 &= 1 \end{align} , \mu_1 = 2 , \begin{align}\mu_1 &= 1 \\ \mu_2 &= 1 \end{align} , - ! scope="row" , Geometric ,
\gamma_i = \gamma(\lambda_i) , \gamma_1 = 2 , \begin{align}\gamma_1 &= 1 \\ \gamma_2 &= 1 \end{align} , \begin{align}\gamma_1 &= 1 \\ \gamma_2 &= 1 \end{align} , \gamma_1 = 1 , \begin{align}\gamma_1 &= 1 \\ \gamma_2 &= 1 \end{align} , - ! scope="row" , Eigenvectors , All nonzero vectors , \begin{align} \mathbf u_1 &= \begin{bmatrix} 1\\ 0\end{bmatrix} \\ \mathbf u_2 &= \begin{bmatrix} 0\\ 1\end{bmatrix} \end{align} , \begin{align} \mathbf u_1 &= \begin{bmatrix} 1\\ -i\end{bmatrix} \\ \mathbf u_2 &= \begin{bmatrix} 1\\ +i\end{bmatrix} \end{align} , \mathbf u_1 = \begin{bmatrix} 1\\ 0 \end{bmatrix} , \begin{align} \mathbf u_1 &= \begin{bmatrix} 1\\ 1\end{bmatrix} \\ \mathbf u_2 &= \begin{bmatrix} 1\\ -1\end{bmatrix} \end{align} The characteristic equation for a rotation is a
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
with
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the orig ...
D = -4(\sin\theta)^2, which is a negative number whenever is not an integer multiple of 180°. Therefore, except for these special cases, the two eigenvalues are complex numbers, \cos\theta \pm i\sin\theta; and all eigenvectors have non-real entries. Indeed, except for those special cases, a rotation changes the direction of every nonzero vector in the plane. A linear transformation that takes a square to a rectangle of the same area (a squeeze mapping) has reciprocal eigenvalues.


Schrödinger equation

An example of an eigenvalue equation where the transformation T is represented in terms of a differential operator is the time-independent
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
: : H\psi_E = E\psi_E \, where H, the Hamiltonian, is a second-order
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
and \psi_E, the
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
, is one of its eigenfunctions corresponding to the eigenvalue E, interpreted as its
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
. However, in the case where one is interested only in the bound state solutions of the Schrödinger equation, one looks for \psi_E within the space of square integrable functions. Since this space is a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
with a well-defined
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
, one can introduce a basis set in which \psi_E and H can be represented as a one-dimensional array (i.e., a vector) and a matrix respectively. This allows one to represent the Schrödinger equation in a matrix form. The
bra–ket notation In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathem ...
is often used in this context. A vector, which represents a state of the system, in the Hilbert space of square integrable functions is represented by , \Psi_E\rangle. In this notation, the Schrödinger equation is: : H, \Psi_E\rangle = E, \Psi_E\rangle where , \Psi_E\rangle is an eigenstate of H and E represents the eigenvalue. H is an
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phy ...
self-adjoint operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to its ...
, the infinite-dimensional analog of Hermitian matrices. As in the matrix case, in the equation above H, \Psi_E\rangle is understood to be the vector obtained by application of the transformation H to , \Psi_E\rangle.


Wave transport

Light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 t ...
,
acoustic wave Acoustic waves are a type of energy propagation through a medium by means of adiabatic loading and unloading. Important quantities for describing acoustic waves are acoustic pressure, particle velocity, particle displacement and acoustic intensi ...
s, and
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ra ...
s are randomly
scattered Scattered may refer to: Music * ''Scattered'' (album), a 2010 album by The Handsome Family * "Scattered" (The Kinks song), 1993 * "Scattered", a song by Ace Young * "Scattered", a song by Lauren Jauregui * "Scattered", a song by Green Day from ' ...
numerous times when traversing a static disordered system. Even though multiple scattering repeatedly randomizes the waves, ultimately coherent wave transport through the system is a deterministic process which can be described by a field transmission matrix \mathbf{t}. The eigenvectors of the transmission operator \mathbf{t}^\dagger\mathbf{t} form a set of disorder-specific input wavefronts which enable waves to couple into the disordered system's eigenchannels: the independent pathways waves can travel through the system. The eigenvalues, \tau, of \mathbf{t}^\dagger\mathbf{t} correspond to the intensity transmittance associated with each eigenchannel. One of the remarkable properties of the transmission operator of diffusive systems is their bimodal eigenvalue distribution with \tau_\max = 1 and \tau_\min = 0. Furthermore, one of the striking properties of open eigenchannels, beyond the perfect transmittance, is the statistically robust spatial profile of the eigenchannels.


Molecular orbitals

In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, and in particular in atomic and molecular physics, within the Hartree–Fock theory, the atomic and
molecular orbital In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of find ...
s can be defined by the eigenvectors of the Fock operator. The corresponding eigenvalues are interpreted as ionization potentials via Koopmans' theorem. In this case, the term eigenvector is used in a somewhat more general meaning, since the Fock operator is explicitly dependent on the orbitals and their eigenvalues. Thus, if one wants to underline this aspect, one speaks of nonlinear eigenvalue problems. Such equations are usually solved by an iteration procedure, called in this case
self-consistent field In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...
method. In
quantum chemistry Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
, one often represents the Hartree–Fock equation in a non-
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
basis set. This particular representation is a
generalized eigenvalue problem In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the ...
called
Roothaan equations The Roothaan equations are a representation of the Hartree–Fock equation in a non orthonormal basis set which can be of Gaussian-type or Slater-type. It applies to closed-shell molecules or atoms where all molecular orbitals or atomic orbita ...
.


Geology and glaciology

In
geology Geology () is a branch of natural science concerned with Earth and other Astronomical object, astronomical objects, the features or rock (geology), rocks of which it is composed, and the processes by which they change over time. Modern geology ...
, especially in the study of
glacial till image:Geschiebemergel.JPG, Closeup of glacial till. Note that the larger grains (pebbles and gravel) in the till are completely surrounded by the matrix of finer material (silt and sand), and this characteristic, known as ''matrix support'', is d ...
, eigenvectors and eigenvalues are used as a method by which a mass of information of a clast fabric's constituents' orientation and dip can be summarized in a 3-D space by six numbers. In the field, a geologist may collect such data for hundreds or thousands of clasts in a soil sample, which can only be compared graphically such as in a Tri-Plot (Sneed and Folk) diagram, or as a Stereonet on a Wulff Net. The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space. The three eigenvectors are ordered \mathbf v_1, \mathbf v_2, \mathbf v_3 by their eigenvalues E_1 \geq E_2 \geq E_3; \mathbf v_1 then is the primary orientation/dip of clast, \mathbf v_2 is the secondary and \mathbf v_3 is the tertiary, in terms of strength. The clast orientation is defined as the direction of the eigenvector, on a
compass rose A compass rose, sometimes called a wind rose, rose of the winds or compass star, is a figure on a compass, map, nautical chart, or monument used to display the orientation of the cardinal directions (north, east, south, and west) and thei ...
of 360°. Dip is measured as the eigenvalue, the modulus of the tensor: this is valued from 0° (no dip) to 90° (vertical). The relative values of E_1, E_2, and E_3 are dictated by the nature of the sediment's fabric. If E_1 = E_2 = E_3, the fabric is said to be isotropic. If E_1 = E_2 > E_3, the fabric is said to be planar. If E_1 > E_2 > E_3, the fabric is said to be linear.


Principal component analysis

The eigendecomposition of a symmetric positive semidefinite (PSD)
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix is used in
multivariate analysis Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable. Multivariate statistics concerns understanding the different aims and background of each of the diff ...
, where the
sample Sample or samples may refer to: Base meaning * Sample (statistics), a subset of a population – complete data set * Sample (signal), a digital discrete sample of a continuous analog signal * Sample (material), a specimen or small quantity of ...
covariance matrices In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
are PSD. This orthogonal decomposition is called
principal component analysis Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...
(PCA) in statistics. PCA studies
linear relation In linear algebra, a linear relation, or simply relation, between elements of a vector space or a module is a linear equation that has these elements as a solution. More precisely, if e_1,\dots,e_n are elements of a (left) module over a ring ...
s among variables. PCA is performed on the
covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
or the
correlation matrix In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
(in which each variable is scaled to have its
sample variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
equal to one). For the covariance or correlation matrix, the eigenvectors correspond to
principal components Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...
and the eigenvalues to the variance explained by the principal components. Principal component analysis of the correlation matrix provides an orthogonal basis for the space of the observed data: In this basis, the largest eigenvalues correspond to the principal components that are associated with most of the covariability among a number of observed data. Principal component analysis is used as a means of
dimensionality reduction Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally ...
in the study of large data sets, such as those encountered in
bioinformatics Bioinformatics () is an interdisciplinary field that develops methods and software tools for understanding biological data, in particular when the data sets are large and complex. As an interdisciplinary field of science, bioinformatics combi ...
. In
Q methodology Q methodology is a research method used in psychology and in social sciences to study people's "subjectivity"—that is, their viewpoint. Q was developed by psychologist William Stephenson. It has been used both in clinical settings for assessing a ...
, the eigenvalues of the correlation matrix determine the Q-methodologist's judgment of ''practical'' significance (which differs from the
statistical significance In statistical hypothesis testing, a result has statistical significance when it is very unlikely to have occurred given the null hypothesis (simply by chance alone). More precisely, a study's defined significance level, denoted by \alpha, is the p ...
of
hypothesis testing A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis. Hypothesis testing allows us to make probabilistic statements about population parameters. ...
; cf. criteria for determining the number of factors). More generally, principal component analysis can be used as a method of
factor analysis Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors. For example, it is possible that variations in six observed ...
in structural equation modeling.


Vibration analysis

Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
. The eigenvalues are the natural frequencies (or eigenfrequencies) of vibration, and the eigenvectors are the shapes of these vibrational modes. In particular, undamped vibration is governed by m\ddot{x} + kx = 0 or m\ddot{x} = -kx that is, acceleration is proportional to position (i.e., we expect x to be sinusoidal in time). In n dimensions, m becomes a mass matrix and k a stiffness matrix. Admissible solutions are then a linear combination of solutions to the
generalized eigenvalue problem In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the ...
kx = \omega^2 mx where \omega^2 is the eigenvalue and \omega is the (imaginary)
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
. The principal vibration modes are different from the principal compliance modes, which are the eigenvectors of k alone. Furthermore,
damped vibration Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic function, periodic, such as the motion of a pendulum ...
, governed by m\ddot{x} + c\dot{x} + kx = 0 leads to a so-called quadratic eigenvalue problem, \left(\omega^2 m + \omega c + k\right)x = 0. This can be reduced to a generalized eigenvalue problem by algebraic manipulation at the cost of solving a larger system. The orthogonality properties of the eigenvectors allows decoupling of the differential equations so that the system can be represented as linear summation of the eigenvectors. The eigenvalue problem of complex structures is often solved using
finite element analysis The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
, but neatly generalize the solution to scalar-valued vibration problems.


Eigenfaces

In
image processing An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
, processed images of faces can be seen as vectors whose components are the
brightness Brightness is an attribute of visual perception in which a source appears to be radiating or reflecting light. In other words, brightness is the perception elicited by the luminance of a visual target. The perception is not linear to luminance, ...
es of each
pixel In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device. In most digital display devices, pixels are the ...
. The dimension of this vector space is the number of pixels. The eigenvectors of the
covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
associated with a large set of normalized pictures of faces are called
eigenface An eigenface () is the name given to a set of eigenvectors when used in the computer vision problem of human face recognition. The approach of using eigenfaces for recognition was developed by Sirovich and Kirby and used by Matthew Turk and Ale ...
s; this is an example of
principal component analysis Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...
. They are very useful for expressing any face image as a linear combination of some of them. In the facial recognition branch of
biometrics Biometrics are body measurements and calculations related to human characteristics. Biometric authentication (or realistic authentication) is used in computer science as a form of identification and access control. It is also used to identify i ...
, eigenfaces provide a means of applying
data compression In information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compressio ...
to faces for
identification Identification or identify may refer to: *Identity document, any document used to verify a person's identity Arts, entertainment and media * ''Identify'' (album) by Got7, 2014 * "Identify" (song), by Natalie Imbruglia, 1999 * Identification ( ...
purposes. Research related to eigen vision systems determining hand gestures has also been made. Similar to this concept, eigenvoices represent the general direction of variability in human pronunciations of a particular utterance, such as a word in a language. Based on a linear combination of such eigenvoices, a new voice pronunciation of the word can be constructed. These concepts have been found useful in automatic speech recognition systems for speaker adaptation.


Tensor of moment of inertia

In
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objec ...
, the eigenvectors of the moment of inertia tensor define the principal axes of a
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
. The
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
of moment of
inertia Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
is a key quantity required to determine the rotation of a rigid body around its
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
.


Stress tensor

In
solid mechanics Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and ...
, the
stress Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
tensor is symmetric and so can be decomposed into a
diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ� ...
tensor with the eigenvalues on the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation, the stress tensor has no shear components; the components it does have are the principal components.


Graphs

In
spectral graph theory In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matri ...
, an eigenvalue of a graph is defined as an eigenvalue of the graph's
adjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simp ...
A, or (increasingly) of the graph's
Laplacian matrix In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph La ...
due to its
discrete Laplace operator In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid. For the case of a finite-dimensional graph (having a finite number of edges and vertic ...
, which is either D - A (sometimes called the ''combinatorial Laplacian'') or I - D^{-1/2}A D^{-1/2} (sometimes called the ''normalized Laplacian''), where D is a diagonal matrix with D_{ii} equal to the degree of vertex v_i, and in D^{-1/2}, the ith diagonal entry is 1/\sqrt{\deg(v_i)}. The kth principal eigenvector of a graph is defined as either the eigenvector corresponding to the kth largest or kth smallest eigenvalue of the Laplacian. The first principal eigenvector of the graph is also referred to merely as the principal eigenvector. The principal eigenvector is used to measure the centrality of its vertices. An example is
Google Google LLC () is an American Multinational corporation, multinational technology company focusing on Search Engine, search engine technology, online advertising, cloud computing, software, computer software, quantum computing, e-commerce, ar ...
's
PageRank PageRank (PR) is an algorithm used by Google Search to rank webpages, web pages in their search engine results. It is named after both the term "web page" and co-founder Larry Page. PageRank is a way of measuring the importance of website pages. A ...
algorithm. The principal eigenvector of a modified
adjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simp ...
of the World Wide Web graph gives the page ranks as its components. This vector corresponds to the
stationary distribution Stationary distribution may refer to: * A special distribution for a Markov chain such that if the chain starts with its stationary distribution, the marginal distribution of all states at any time will always be the stationary distribution. Assum ...
of the
Markov chain A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happen ...
represented by the row-normalized adjacency matrix; however, the adjacency matrix must first be modified to ensure a stationary distribution exists. The second smallest eigenvector can be used to partition the graph into clusters, via spectral clustering. Other methods are also available for clustering.


Basic reproduction number

The basic reproduction number (R_0) is a fundamental number in the study of how infectious diseases spread. If one infectious person is put into a population of completely susceptible people, then R_0 is the average number of people that one typical infectious person will infect. The generation time of an infection is the time, t_G, from one person becoming infected to the next person becoming infected. In a heterogeneous population, the next generation matrix defines how many people in the population will become infected after time t_G has passed. R_0 is then the largest eigenvalue of the next generation matrix.


See also

* Antieigenvalue theory * Eigenoperator * Eigenplane * Eigenmoments * Eigenvalue algorithm * Introduction to eigenstates * Jordan normal form *
List of numerical-analysis software Listed here are notable end-user computer applications intended for use with numerical or data analysis: Numerical-software packages General-purpose computer algebra systems Interface-oriented Language-oriented Historically signific ...
* Nonlinear eigenproblem * Normal eigenvalue * Quadratic eigenvalue problem * Singular value *
Spectrum of a matrix In mathematics, the spectrum of a matrix is the set of its eigenvalues. More generally, if T\colon V \to V is a linear operator on any finite-dimensional vector space, its spectrum is the set of scalars \lambda such that T-\lambda I is not invert ...


Notes


Citations


Sources

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *


Further reading

* * * * *


External links


What are Eigen Values?
– non-technical introduction from PhysLink.com's "Ask the Experts"

– Tutorial and Interactive Program from Revoledu.
Introduction to Eigen Vectors and Eigen Values
– lecture from Khan Academy
Eigenvectors and eigenvalues , Essence of linear algebra, chapter 10
– A visual explanation with 3Blue1Brown
Matrix Eigenvectors Calculator
from Symbolab (Click on the bottom right button of the 2x12 grid to select a matrix size. Select an n \times n size (for a square matrix), then fill out the entries numerically and click on the Go button. It can accept complex numbers as well.)


Theory





Edited by Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe, and
Henk van der Vorst Hendrik "Henk" Albertus van der Vorst (born 5 May 1944, Venlo) is a Dutch mathematician and Emeritus Professor of Numerical Analysis at Utrecht University. According to the Institute for Scientific Information (ISI), his paper on the BiCGST ...
{{DEFAULTSORT:Eigenvalues And Eigenvectors Abstract algebra Linear algebra Mathematical physics Matrix theory Singular value decomposition