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 ...
, a Hermitian matrix (or self-adjoint matrix) is a
complex 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 ...
that is equal to its own
conjugate transpose—that is, the element in the -th row and -th column is equal to the
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 - ...
of the element in the -th row and -th column, for all indices and :
or in matrix form:
Hermitian matrices can be understood as the complex extension of real
symmetric matrices.
If the
conjugate transpose of a matrix
is denoted by
, then the Hermitian property can be written concisely as
Hermitian matrices are named after
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.
...
, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real
eigenvalues
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 ...
. Other, equivalent notations in common use are
, although note that 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, ...
,
typically means the
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 - ...
only, and not the
conjugate transpose.
Alternative characterizations
Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below:
Equality with the adjoint
A square matrix
is Hermitian if and only if it is equal to its
adjoint, that is, it satisfies
for any pair of vectors
, where
denotes
the inner product operation.
This is also the way that the more general concept of
self-adjoint operator is defined.
Reality of quadratic forms
An
matrix
is Hermitian if and only if
Spectral properties
A square matrix
is Hermitian if and only if it is unitarily
diagonalizable with real
eigenvalues
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 ...
.
Applications
Hermitian matrices are fundamental to
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, ...
because they describe operators with necessarily real eigenvalues. An eigenvalue
of an operator
on some quantum state
is one of the possible measurement outcomes of the operator, which necessitates the need for operators with real eigenvalues.
Examples and solutions
In this section, the conjugate transpose of matrix
is denoted as
the transpose of matrix
is denoted as
and conjugate of matrix
is denoted as
See the following example:
The diagonal elements must be
real, as they must be their own complex conjugate.
Well-known families of Hermitian matrices include the
Pauli matrices, the
Gell-Mann matrices and their generalizations. In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
such Hermitian matrices are often multiplied by
imaginary coefficients, which results in
skew-Hermitian matrices.
Here, we offer another useful Hermitian matrix using an abstract example. If a square matrix
equals the
multiplication of a matrix and its conjugate transpose, that is,
then
is a Hermitian
positive semi-definite matrix. Furthermore, if
is row full-rank, then
is positive definite.
Properties
Main diagonal values are real
The entries on the
main diagonal
In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matri ...
(top left to bottom right) of any Hermitian matrix are
real.
Only the
main diagonal
In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matri ...
entries are necessarily real; Hermitian matrices can have arbitrary complex-valued entries in their
off-diagonal elements, as long as diagonally-opposite entries are complex conjugates.
Symmetric
A matrix that has only real entries is
symmetric 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 ...
it is Hermitian matrix. A real and symmetric matrix is simply a special case of a Hermitian matrix.
So, if a real anti-symmetric matrix is multiplied by a multiple of imaginary unit
then it becomes Hermitian.
Normal
Every Hermitian matrix is a
normal matrix. That is to say,
Diagonalizable
The finite-dimensional
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful be ...
says that any Hermitian matrix can be
diagonalized by a
unitary matrix, and that the resulting diagonal matrix has only real entries. This implies that all
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 of a Hermitian matrix with dimension are real, and that has linearly independent
eigenvectors. Moreover, a Hermitian matrix has
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 ...
eigenvectors for distinct eigenvalues. Even if there are degenerate eigenvalues, it is always possible to find an
orthogonal basis of consisting of eigenvectors of .
Sum of Hermitian matrices
The sum of any two Hermitian matrices is Hermitian.
Inverse is Hermitian
The
inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when a ...
of an invertible Hermitian matrix is Hermitian as well.
Associative product of Hermitian matrices
The
product of two Hermitian matrices and is Hermitian if and only if .
''ABA'' Hermitian
If ''A'' and ''B'' are Hermitian, then ''ABA'' is also Hermitian.
is real for complex
For an arbitrary complex valued vector the product
is real because of
This is especially important in quantum physics where Hermitian matrices are operators that measure properties of a system e.g. total
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally ...
which have to be real.
Complex Hermitian forms vector space over
The Hermitian complex -by- matrices do not form a
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 the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, , since the identity matrix is Hermitian, but is not. However the complex Hermitian matrices ''do'' form a vector space over the
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every re ...
. In the -
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
vector space of complex matrices over , the complex Hermitian matrices form a subspace of dimension . If denotes the -by- matrix with a in the position and zeros elsewhere, a basis (orthonormal with respect to the Frobenius inner product) can be described as follows:
together with the set of matrices of the form
and the matrices
where
denotes the
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 ...
,
An example is that the four
Pauli matrices form a complete basis for the vector space of all complex 2-by-2 Hermitian matrices over .
Eigendecomposition
If orthonormal eigenvectors
of a Hermitian matrix are chosen and written as the columns of the matrix , then one
eigendecomposition of is
where
and therefore
where
are the eigenvalues on the diagonal of the diagonal matrix
Real determinant
The determinant of a Hermitian matrix is real:
(Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.)
Decomposition into Hermitian and skew-Hermitian matrices
Additional facts related to Hermitian matrices include:
* The sum of a square matrix and its conjugate transpose
is Hermitian.
* The difference of a square matrix and its conjugate transpose
is
skew-Hermitian (also called antihermitian). This implies that the
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
of two Hermitian matrices is skew-Hermitian.
* An arbitrary square matrix can be written as the sum of a Hermitian matrix and a skew-Hermitian matrix . This is known as the Toeplitz decomposition of .
Rayleigh quotient
In mathematics, for a given complex Hermitian matrix and nonzero vector , the Rayleigh quotient
, is defined as:
[Parlet B. N. ''The symmetric eigenvalue problem'', SIAM, Classics in Applied Mathematics,1998]
For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose
to the usual transpose
. Note that
for any non-zero real scalar
. Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues.
It can be shown that, for a given matrix, the Rayleigh quotient reaches its minimum value
(the smallest eigenvalue of M) when
is
(the corresponding eigenvector). Similarly,
and
The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration.
The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range (or spectrum in functional analysis). When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis,
is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to associates the Rayleigh quotient for a fixed and varying through the algebra would be referred to as "vector state" of the algebra.
See also
*
Complex symmetric matrix
*
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 ...
*
Skew-Hermitian matrix (anti-Hermitian matrix)
*
Haynsworth inertia additivity formula
*
Hermitian form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allow ...
*
Self-adjoint operator
*
Unitary matrix
*
Normal matrix
References
External links
*
Visualizing Hermitian Matrix as An Ellipse with Dr. Geo by Chao-Kuei Hung from Chaoyang University, gives a more geometric explanation.
*
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Matrices