Hermitian matrix

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https://en.wikipedia.org/wiki/Hermitian_matrix

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j:

${\displaystyle A{\text{ Hermitian}}\quad \iff \quad a_{ij}={\overline {{a}_{ji}}}}$

or in matrix form:

$${\displaystyle A{\text{ Hermitian}}\quad \iff \quad A={\overline {A^{\mathsf {T}}}}.}$$

Hermitian matrices can be understood as the complex extension of real symmetric matrices.

If the conjugate transpose of a matrix ${\displaystyle A}$ is denoted by ${\displaystyle A^{\mathsf {H}}}$, then the Hermitian property can be written concisely as

${\displaystyle A{\text{ Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}}$

Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are ${\displaystyle A^{\mathsf {H}}=A^{\dagger }=A^{\ast }}$, although note that in quantum mechanics, ${\displaystyle A^{\ast }}$ typically means the complex conjugate 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 ${\displaystyle A}$ is Hermitian if and only if it is equal to its adjoint, that is, it satisfies

$${\displaystyle \langle \mathbf {w} ,A\mathbf {v} \rangle =\langle A\mathbf {w} ,\mathbf {v} \rangle ,}$$

for any pair of vectors ${\displaystyle \mathbf {v} ,\mathbf {w} }$, where ${\displaystyle \langle \cdot ,\cdot \rangle }$ 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

A square matrix ${\displaystyle A}$ is Hermitian if and only if

$${\displaystyle \langle \mathbf {v} ,A\mathbf {v} \rangle \in \mathbb {R} ,\quad \mathbf {v} \in \mathbb {R} .}$$

Spectral properties

A square matrix ${\displaystyle A}$ is Hermitian if and only if it is unitarily diagonalizable with real eigenvalues.

Applications

Hermitian matrices are fundamental to the quantum theory of matrix mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.

Examples and solutions

In this section, the conjugate transpose of matrix ${\displaystyle A}$ is denoted as ${\displaystyle A^{\mathsf {H}},}$ the transpose of matrix ${\displaystyle A}$ is denoted as ${\displaystyle A^{\mathsf {T}}}$ and conjugate of matrix ${\displaystyle A}$ is denoted as ${\displaystyle {\overline {A}}.}$

See the following example:

$${\displaystyle {\begin{bmatrix}0&a-ib&c-id\\a+ib&1&m-in\\c+id&m+in&2\end{bmatrix}}}$$

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 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 ${\displaystyle A}$ equals the multiplication of a matrix and its conjugate transpose, that is, ${\displaystyle A=BB^{\mathsf {H}},}$ then ${\displaystyle A}$ is a Hermitian positive semi-definite matrix. Furthermore, if ${\displaystyle B}$ is row full-rank, then ${\displaystyle A}$ is positive definite.

Properties

Main diagonal values are real

The entries on the main diagonal (top left to bottom right) of any Hermitian matrix are real.

Proof

By definition of the Hermitian matrix

$${\displaystyle H_{ij}={\overline {H}}_{ji}}$$

so for i = j the above follows.

Only the main diagonal 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 it is Hermitian matrix. A real and symmetric matrix is simply a special case of a Hermitian matrix.

Proof

${\displaystyle H_{ij}={\overline {H}}_{ji}}$ by definition. Thus ${\displaystyle H_{ij}=H_{ji}}$ (matrix symmetry) if and only if ${\displaystyle H_{ij}={\overline {H}}_{ij}}$ (${\displaystyle H_{ij}}$ is real).

So, if a real anti-symmetric matrix is multiplied by a multiple of imaginary unit ${\displaystyle i,}$ then it becomes Hermitian.

Normal

Every Hermitian matrix is a normal matrix. That is to say, ${\displaystyle AA^{\mathsf {H}}=A^{\mathsf {H}}A.}$

Proof

${\displaystyle A=A^{\mathsf {H}}}$, so ${\displaystyle AA^{\mathsf {H}}=AA=A^{\mathsf {H}}A.}$

Diagonalizable

The finite-dimensional spectral theorem 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 eigenvalues of a Hermitian matrix A with dimension n are real, and that A has n linearly independent eigenvectors. Moreover, a Hermitian matrix has orthogonal eigenvectors for distinct eigenvalues. Even if there are degenerate eigenvalues, it is always possible to find an orthogonal basis of Cⁿ consisting of n eigenvectors of A.

Sum of Hermitian matrices

The sum of any two Hermitian matrices is Hermitian.

Proof

$${\displaystyle (A+B)_{ij}=A_{ij}+B_{ij}={\overline {A}}_{ji}+{\overline {B}}_{ji}={\overline {(A+B)}}_{ji},}$$

as claimed.

Inverse is Hermitian

The inverse of an invertible Hermitian matrix is Hermitian as well.

Proof

If ${\displaystyle A^{-1}A=I}$, then ${\displaystyle I=I^{\mathsf {H}}=\left(A^{-1}A\right)^{\mathsf {H}}=A^{\mathsf {H}}\left(A^{-1}\right)^{\mathsf {H}}=A\left(A^{-1}\right)^{\mathsf {H}}}$, so ${\displaystyle A^{-1}=\left(A^{-1}\right)^{\mathsf {H}}}$ as claimed.

Associative product of Hermitian matrices

The product of two Hermitian matrices A and B is Hermitian if and only if AB = BA.

Proof

Note that

$${\displaystyle (AB)^{\mathsf {H}}={\overline {(AB)^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}A^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}}}\ {\overline {A^{\mathsf {T}}}}=B^{\mathsf {H}}A^{\mathsf {H}}=BA.}$$

Thus ${\displaystyle (AB)^{\mathsf {H}}=AB}$ if and only if ${\displaystyle AB=BA.}$

Thus Aⁿ is Hermitian if A is Hermitian and n is an integer.

ABA Hermitian

If A and B are Hermitian, then ABA is also Hermitian.

Proof

$${\displaystyle (ABA)^{\mathsf {H}}=(A(BA))^{\mathsf {H}}=(BA)^{\mathsf {H}}A^{\mathsf {H}}=A^{\mathsf {H}}B^{\mathsf {H}}A^{\mathsf {H}}=ABA}$$

v^HAv is real for complex v

For an arbitrary complex valued vector v the product ${\displaystyle \mathbf {v} ^{\mathsf {H}}A\mathbf {v} }$ is real because of ${\displaystyle \mathbf {v} ^{\mathsf {H}}A\mathbf {v} =\left(\mathbf {v} ^{\mathsf {H}}A\mathbf {v} \right)^{\mathsf {H}}.}$ This is especially important in quantum physics where Hermitian matrices are operators that measure properties of a system e.g. total spin which have to be real.

Complex Hermitian forms vector space over R

The Hermitian complex n-by-n matrices do not form a vector space over the complex numbers, C, since the identity matrix I_n is Hermitian, but i I_n is not. However the complex Hermitian matrices do form a vector space over the real numbers R. In the 2n²-dimensional vector space of complex n × n matrices over R, the complex Hermitian matrices form a subspace of dimension n². If E_jk denotes the n-by-n matrix with a 1 in the j,k position and zeros elsewhere, a basis (orthonormal with respect to the Frobenius inner product) can be described as follows:

$${\displaystyle E_{jj}{\text{ for }}1\leq j\leq n\quad (n{\text{ matrices}})}$$

together with the set of matrices of the form

$${\displaystyle {\frac {1}{\sqrt {2}}}\left(E_{jk}+E_{kj}\right){\text{ for }}1\leq j<k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}$$

and the matrices

$${\displaystyle {\frac {i}{\sqrt {2}}}\left(E_{jk}-E_{kj}\right){\text{ for }}1\leq j<k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}$$

where ${\displaystyle i}$ denotes the imaginary unit, ${\displaystyle i={\sqrt {-1}}~.}$

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 R.

Eigendecomposition

If n orthonormal eigenvectors ${\displaystyle \mathbf {u} _{1},\dots ,\mathbf {u} _{n}}$ of a Hermitian matrix are chosen and written as the columns of the matrix U, then one eigendecomposition of A is ${\displaystyle A=U\Lambda U^{\mathsf {H}}}$ where ${\displaystyle UU^{\mathsf {H}}=I=U^{\mathsf {H}}U}$ and therefore

$${\displaystyle A=\sum _{j}\lambda _{j}\mathbf {u} _{j}\mathbf {u} _{j}^{\mathsf {H}},}$$

where ${\displaystyle \lambda _{j}}$ are the eigenvalues on the diagonal of the diagonal matrix ${\displaystyle \Lambda .}$

Real determinant

The determinant of a Hermitian matrix is real:

Proof

${\displaystyle \det(A)=\det \left(A^{\mathsf {T}}\right)\quad \Rightarrow \quad \det \left(A^{\mathsf {H}}\right)={\overline {\det(A)}}}$ Therefore if ${\displaystyle A=A^{\mathsf {H}}\quad \Rightarrow \quad \det(A)={\overline {\det(A)}}}$.

(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 ${\displaystyle \left(A+A^{\mathsf {H}}\right)}$ is Hermitian.
• The difference of a square matrix and its conjugate transpose ${\displaystyle \left(A-A^{\mathsf {H}}\right)}$ is skew-Hermitian (also called antihermitian). This implies that the commutator of two Hermitian matrices is skew-Hermitian.
• An arbitrary square matrix C can be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B. This is known as the Toeplitz decomposition of C.
  $${\displaystyle C=A+B\quad {\text{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\text{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right)}$$

  

Rayleigh quotient

Main article: Rayleigh quotient

In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient ${\displaystyle R(M,\mathbf {x} )}$, is defined as:

$${\displaystyle R(M,\mathbf {x} ):={\frac {\mathbf {x} ^{\mathsf {H}}M\mathbf {x} }{\mathbf {x} ^{\mathsf {H}}\mathbf {x} }}.}$$

For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose ${\displaystyle \mathbf {x} ^{\mathsf {H}}}$ to the usual transpose ${\displaystyle \mathbf {x} ^{\mathsf {T}}}$. Note that ${\displaystyle R(M,c\mathbf {x} )=R(M,\mathbf {x} )}$ for any non-zero real scalar ${\displaystyle c}$. 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 ${\displaystyle \lambda _{\min }}$ (the smallest eigenvalue of M) when ${\displaystyle \mathbf {x} }$ is ${\displaystyle \mathbf {v} _{\min }}$ (the corresponding eigenvector). Similarly, ${\displaystyle R(M,\mathbf {x} )\leq \lambda _{\max }}$ and ${\displaystyle R(M,\mathbf {v} _{\max })=\lambda _{\max }.}$

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, ${\displaystyle \lambda _{\max }}$ is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to M associates the Rayleigh quotient R(M, x) for a fixed x and M varying through the algebra would be referred to as "vector state" of the algebra.