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Hermitian matrix

Let $\;M_n$ be the set of $\;n \times n$ complex-valued matrices. Let us consider a matrix $\; A=[a_{ij}] \in M_n$ and denote its complex conjugate by $\; \overline{A}=[\overline{a}_{ij}]$ and its transpose by $\; A^T=[a_{ji}]$ . We then have the following

Definition: A matrix $\; A=[a_{ij}] \in M_n$ is said to be Hermitian if $\; A=A^*$ , where $\; A^*=\overline{A}^T=[\overline{a}_{ji}]$ . It is skew-Hermitian if $\; A=-A^*$ .

A Hermitian matrix can be the representation, in a given orthonormal basis, of a self-adjoint operator.

Properties of Hermitian matrices

For two matrices $\; A, B \in M_n$ we have:

If $\; A$ is Hermitian, then the main diagonal entries of $\; A$ are all real. In order to specify the $\; n^2$ elements of $\; A$ one may specify freely any $\; n$ real numbers for the main diagonal entries and any $\; \frac{1}{2}n(n-1)$ complex numbers for the off-diagonal entries;
$\; A+A^*$ , $\; AA^*$ and $\; A^*A$ are all Hermitian for all $\; A \in M_n$ ;
If $\; A$ is Hermitian, then $\; A^k$ is Hermitian for all $\; k=1, 2, 3, \ldots$ . If $\; A$ is nonsingular as well, then $\; A^{-1}$ is Hermitian;
If $\; A, B$ are Hermitian, then $\; aA+bB$ is Hermitian for all real scalars $\; a, b$ ;
$\; A-A^*$ is skew-Hermitian for all $\; A \in M_n$ ;
If $\; A, B$ are skew-Hermitian, then $\; aA+bB$ is skew-Hermitian for all real scalars $\; a, b$ ;
If $\; A$ is Hermitian, then $\; iA$ is skew-Hermitian;
If $\; A$ is skew-Hermitian, then $\; iA$ is Hermitian;
Any $\; A \in M_n$ can be written as

$\; A=\frac{1}{2}(A+A^*)+\frac{1}{2}(A-A^*)\equiv H(A)+S(A) ,$

where $\; H(A)=\frac{1}{2}(A+A^*)$ respectively $\; S(A)= \frac{1}{2}(A-A^*)$ are the Hermitian and skew-Hermitian parts of $\; A$ .

Theorem: Each $\; A \in M_n$ can be written uniquely as $\; A = S + iT$ , where $\; S$ and $\; T$ are both Hermitian. It can also be written uniquely as $\; A = B + C$ , where $\; B$ is Hermitian and $\; C$ is skew-Hermitian.

Theorem: Let $\; A \in M_n$ be Hermitian. Then

$\; x^*Ax$ is real for all $\; x \in \mathbf{C}^n$ ;
All the eigenvalues of $\; A$ are real; and
$\; S^*AS$ is Hermitian for all $\; S \in M_n$ .

Theorem: Let $\; A=[a_{ij}] \in M_n$ be given. Then $\; A$ is Hermitian if and only if at least one of the following holds:

$\; x^*Ax$ is real for all $\; x \in \mathbf{C}^n$ ;
$\; A$ is normal and all the eigenvalues of $\; A$ are real; or
$\; S^*AS$ is Hermitian for all $\; S \in M_n$ .

Theorem [the spectral theorem for Hermitian matrices]: Let $\; A=[a_{ij}] \in M_n$ be given. Then $\; A$ is Hermitian if and only if there are a unitary matrix $\; U \in M_n$ and a real diagonal matrix $\; \Lambda \in M_n$ such that $\; A=U\Lambda U^*$ . Moreover, $\; A$ is real and Hermitian (i.e. real symmetric) if and only if there exist a real orthogonal matrix $\; P \in M_n$ and a real diagonal matrix $\; \Lambda \in M_n$ such that $\; A=P\Lambda P^T$ .

Theorem: Let $\; \mathcal{F}$ be a given family of Hermitian matrices. Then there exists a unitary matrix $\; U \in M_n$ such that $\; U\Lambda U^*$ is diagonal for all $\; A \in \mathcal{F}$ if and only if $\; AB=BA$ for all $\; A, B \in \mathcal{F}$ .

Positivity of Hermitian matrices

Definition: An $\;n \times n$ Hermitian matrix $\; A$ is said to be positive definite if

$\; x^*Ax > 0$ for all $\; x \in \mathbf{C}^n.$

If $\; x^*Ax \geq 0$ , then $\; A$ is said to be positive semidefinite.

The following two theorems give useful and simple characterizations of the positivity of Hermitian matrices.

Theorem: A Hermitian matrix $\; A \in M_n$ is positive semidefinite if and only if all of its eigenvalues are nonnegative. It is positive definite if and only if all of its eigenvalues are positive.

In the following we denote by $\; A_i$ the leading principal submatrix of $\; A$ determined by the first $\; i$ rows and columns: $\; A_i \equiv A(\{1, 2, \ldots, i\}), \; i=2, \ldots, n$ .

As for any positive matrix, if $\; A$ is positive definite, then all principal minors of $\; A$ are positive; when $\; A$ is Hermitian, the converse is also valid. However, an even stronger statement can be made.

Theorem: If $\; A \in M_n$ is Hermitian, then $\; A$ is positive definite if and only if $\; Det A_i>0$ for $\; i=2, \ldots, n$ . More generally, the positivity of any nested sequence of $\; n$ principal minors of $\; A$ is a necessary and sufficient condition for $\; A$ to be positive definite.