In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the spectrum of a
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 ...
is 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 ...
of its
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s. More generally, if
is a
linear operator on any
finite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...
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 ...
, its spectrum is the set of scalars
such that
is not
invertible. 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 equals the product of its eigenvalues. Similarly, the
trace of the matrix equals the sum of its eigenvalues.
From this point of view, we can define the
pseudo-determinant for a
singular matrix to be the product of its nonzero eigenvalues (the density of
multivariate normal distribution will need this quantity).
In many applications, such as
PageRank, one is interested in the dominant eigenvalue, i.e. that which is largest in
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
. In other applications, the smallest eigenvalue is important, but in general, the whole spectrum provides valuable information about a matrix.
Definition
Let ''V'' be a finite-dimensional
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 ...
over some
field ''K'' and suppose ''T'' : ''V'' → ''V'' is a linear map. The ''spectrum'' of ''T'', denoted σ
''T'', is the
multiset of
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
of 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 ''T''. Thus the elements of the spectrum are precisely the eigenvalues of ''T'', and the multiplicity of an eigenvalue ''λ'' in the spectrum equals the dimension of the
generalized eigenspace of ''T'' for ''λ'' (also called the
algebraic multiplicity of ''λ'').
Now, fix a
basis ''B'' of ''V'' over ''K'' and suppose ''M'' ∈ Mat
''K'' (''V'') is a matrix. Define the linear map ''T'' : ''V'' → ''V'' pointwise by ''Tx'' = ''Mx'', where on the right-hand side ''x'' is interpreted as a column vector and ''M'' acts on ''x'' by
matrix multiplication. We now say that ''x'' ∈ ''V'' is an
eigenvector of ''M'' if ''x'' is an eigenvector of ''T''. Similarly, λ ∈ ''K'' is an eigenvalue of ''M'' if it is an eigenvalue of ''T'', and with the same multiplicity, and the spectrum of ''M'', written σ
''M'', is the multiset of all such eigenvalues.
Related notions
The
eigendecomposition (or spectral decomposition) of a
diagonalizable matrix
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. ...
is a
decomposition
Decomposition or rot is the process by which dead organic substances are broken down into simpler organic or inorganic matter such as carbon dioxide, water, simple sugars and mineral salts. The process is a part of the nutrient cycle and ...
of a diagonalizable matrix into a specific canonical form whereby the matrix is represented in terms of its eigenvalues and eigenvectors.
The
spectral radius of 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 ...
is the largest absolute value of its eigenvalues. In
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
, the spectral radius of a
bounded linear operator is the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
of the absolute values of the elements in the spectrum of that operator.
Notes
References
*
*
*
*
Matrix theory
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