Hermitian matrix

Let $\backslash ;M\_n$ be the set of $\backslash ;n\; \backslash times\; n$ complex-valued matrices. Let us consider a matrix $\backslash ;\; A=[a\_\{ij\}]\; \backslash in\; M\_n$ and denote its complex conjugate by $\backslash ;\; \backslash overline\{A\}=[\backslash overline\{a\}\_\{ij\}]$ and its transpose by $\backslash ;\; A^T=[a\_\{ji\}]$. We then have the following '''Definition: A matrix $\backslash ;\; A=[a\_\{ij\}]\; \backslash in\; M\_n$ is said to be ''Hermitian'' if $\backslash ;\; A=A^*$, where $\backslash ;\; A^*=\backslash overline\{A\}^T=[\backslash overline\{a\}\_\{ji\}]$. It is ''skew-Hermitian'' if $\backslash ;\; A=-A^*$.''' A '''Hermitian matrix''' can be the representation, in a given orthonormal basis, of a [[self-adjoint operator]]. == Properties of Hermitian matrices == For two matrices $\backslash ;\; A,\; B\; \backslash in\; M\_n$ we have: # If $\backslash ;\; A$ is Hermitian, then the main diagonal entries of $\backslash ;\; A$ are all real. In order to specify the $\backslash ;\; n^2$ elements of $\backslash ;\; A$ one may specify freely any $\backslash ;\; n$ real numbers for the main diagonal entries and any $\backslash ;\; \backslash frac\{1\}\{2\}n(n-1)$ complex numbers for the off-diagonal entries; # $\backslash ;\; A+A^*$, $\backslash ;\; AA^*$ and $\backslash ;\; A^*A$ are all Hermitian for all $\backslash ;\; A\; \backslash in\; M\_n$; # If $\backslash ;\; A$ is Hermitian, then $\backslash ;\; A^k$ is Hermitian for all $\backslash ;\; k=1,\; 2,\; 3,\; \backslash ldots$. If $\backslash ;\; A$ is nonsingular as well, then $\backslash ;\; A^\{-1\}$ is Hermitian; # If $\backslash ;\; A,\; B$ are Hermitian, then $\backslash ;\; aA+bB$ is Hermitian for all real scalars $\backslash ;\; a,\; b$; # $\backslash ;\; A-A^*$ is skew-Hermitian for all $\backslash ;\; A\; \backslash in\; M\_n$; # If $\backslash ;\; A,\; B$ are skew-Hermitian, then $\backslash ;\; aA+bB$ is skew-Hermitian for all real scalars $\backslash ;\; a,\; b$; # If $\backslash ;\; A$ is Hermitian, then $\backslash ;\; iA$ is skew-Hermitian; # If $\backslash ;\; A$ is skew-Hermitian, then $\backslash ;\; iA$ is Hermitian; # Any $\backslash ;\; A\; \backslash in\; M\_n$ can be written as :$\backslash ;\; A=\backslash frac\{1\}\{2\}(A+A^*)+\backslash frac\{1\}\{2\}(A-A^*)\backslash equiv\; H(A)+S(A)\; ,$ where $\backslash ;\; H(A)=\backslash frac\{1\}\{2\}(A+A^*)$ respectively $\backslash ;\; S(A)=\; \backslash frac\{1\}\{2\}(A-A^*)$ are the Hermitian and skew-Hermitian parts of $\backslash ;\; A$ . '''Theorem: ''' Each $\backslash ;\; A\; \backslash in\; M\_n$ can be written uniquely as $\backslash ;\; A\; =\; S\; +\; iT$, where $\backslash ;\; S$ and $\backslash ;\; T$ are both Hermitian. It can also be written uniquely as $\backslash ;\; A\; =\; B\; +\; C$, where $\backslash ;\; B$ is Hermitian and $\backslash ;\; C$ is skew-Hermitian. '''Theorem:''' Let $\backslash ;\; A\; \backslash in\; M\_n$ be Hermitian. Then # $\backslash ;\; x^*Ax$ is real for all $\backslash ;\; x\; \backslash in\; \backslash mathbf\{C\}^n$; # All the eigenvalues of $\backslash ;\; A$ are real; and # $\backslash ;\; S^*AS$ is Hermitian for all $\backslash ;\; S\; \backslash in\; M\_n$. '''Theorem:''' Let $\backslash ;\; A=[a\_\{ij\}]\; \backslash in\; M\_n$ be given. Then $\backslash ;\; A$ is ''Hermitian'' if and only if at least one of the following holds: # $\backslash ;\; x^*Ax$ is real for all $\backslash ;\; x\; \backslash in\; \backslash mathbf\{C\}^n$; # $\backslash ;\; A$ is normal and all the eigenvalues of $\backslash ;\; A$ are real; or # $\backslash ;\; S^*AS$ is Hermitian for all $\backslash ;\; S\; \backslash in\; M\_n$. '''Theorem [the spectral theorem for Hermitian matrices]: ''' Let $\backslash ;\; A=[a\_\{ij\}]\; \backslash in\; M\_n$ be given. Then $\backslash ;\; A$ is ''Hermitian'' if and only if there are a unitary matrix $\backslash ;\; U\; \backslash in\; M\_n$ and a real diagonal matrix $\backslash ;\; \backslash Lambda\; \backslash in\; M\_n$ such that $\backslash ;\; A=U\backslash Lambda\; U^*$. Moreover, $\backslash ;\; A$ is real and Hermitian (i.e. real symmetric) if and only if there exist a real orthogonal matrix $\backslash ;\; P\; \backslash in\; M\_n$ and a real diagonal matrix $\backslash ;\; \backslash Lambda\; \backslash in\; M\_n$ such that $\backslash ;\; A=P\backslash Lambda\; P^T$. '''Theorem:''' Let $\backslash ;\; \backslash mathcal\{F\}$ be a given family of Hermitian matrices. Then there exists a unitary matrix $\backslash ;\; U\; \backslash in\; M\_n$ such that $\backslash ;\; U\backslash Lambda\; U^*$ is diagonal for all $\backslash ;\; A\; \backslash in\; \backslash mathcal\{F\}$ if and only if $\backslash ;\; AB=BA$ for all $\backslash ;\; A,\; B\; \backslash in\; \backslash mathcal\{F\}$. == Positivity of Hermitian matrices == '''Definition:''' An $\backslash ;n\; \backslash times\; n$ Hermitian matrix $\backslash ;\; A$ is said to be ''positive definite'' if :$\backslash ;\; x^*Ax\; >\; 0$ for all $\backslash ;\; x\; \backslash in\; \backslash mathbf\{C\}^n.$ If $\backslash ;\; x^*Ax\; \backslash geq\; 0$, then $\backslash ;\; A$ 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 $\backslash ;\; A\; \backslash in\; M\_n$ 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 $\backslash ;\; A\_i$ the leading principal submatrix of $\backslash ;\; A$ determined by the first $\backslash ;\; i$ rows and columns: $\backslash ;\; A\_i\; \backslash equiv\; A(\backslash \{1,\; 2,\; \backslash ldots,\; i\backslash \}),\; \backslash ;\; i=2,\; \backslash ldots,\; n$. As for any positive matrix, if $\backslash ;\; A$ is positive definite, then ''all'' principal minors of $\backslash ;\; A$ are positive; when $\backslash ;\; A$ is ''Hermitian'', the converse is also valid. However, an even stronger statement can be made. '''Theorem:''' If $\backslash ;\; A\; \backslash in\; M\_n$ is Hermitian, then $\backslash ;\; A$ is positive definite if and only if $\backslash ;\; Det\; A\_i>0$ for $\backslash ;\; i=2,\; \backslash ldots,\; n$. More generally, the positivity of ''any'' nested sequence of $\backslash ;\; n$ principal minors of $\backslash ;\; A$ is a necessary and sufficient condition for $\backslash ;\; A$ to be positive definite. [[Category:Linear Algebra]] [[category:Handbook of Quantum Information]] == Bibliography == * R. A. Horn and C. R. Johnson, ''Matrix analysis'', Cambridge University Press (1985).