| Version 9 |
Version 8 |
| For a matrix $A$, let $A^\ast=\overline{A}^{T}$, where |
For a matrix $A$, let $A^\ast=\overline{A}^{T}$, where |
| $A^T$ is the transpose, and $\bar{A}$ is the complex conjugate of $A$. |
$A^T$ is the transpose, and $\bar{A}$ is the complex conjugate of $A$. |
|
|
| {\bf Definition} |
{\bf Definition} |
| A complex square matrix $A$ is \emph{Hermitian}, if |
A complex square matrix $A$ is \emph{Hermitian}, if |
| $$ A = A^*. $$ |
$$ A = A^*. $$ |
|
|
| \subsubsection*{Properties} |
\subsubsection*{Properties} |
| \begin{enumerate} |
\begin{enumerate} |
| \item The eigenvalues of a Hermitian matrix are real \cite{eves}. |
\item The eigenvalues of a Hermitian matrix are real \cite{eves}. |
| \item The diagonal elements of a Hermitian matrix are real. |
\item The diagonal elements of a Hermitian matrix are real. |
| \item The complex conjugate of a Hermitian matrix is a Hermitian matrix. |
\item The complex conjugate of a Hermitian matrix is a Hermitian matrix. |
| \item If $A$ is a Hermitian matrix, and $B$ is a complex matrix |
\item If $A$ is a Hermitian matrix, and $B$ is a complex matrix |
| of same dimension as $A$, then $BAB^\ast$ is a Hermitian matrix. |
of same dimension as $A$, then $BAB^\ast$ is a Hermitian matrix. |
| \item A matrix is symmetric if and only if it is real and Hermitian. |
\item A matrix is symmetric if and only if it is real and Hermitian. |
| \item Hermitian matrices is a vector subspace in the vector space of |
\item Hermitian matrices is a vector subspace in the vector space of |
| complex matrices. |
complex matrices. |
| The real symmetric matrices are a subspace of the Hermitian matrices. |
The real symmetric matrices are a subspace of the Hermitian matrices. |
|
|
| \item Hermitian matrices are also called \emph{self-adjoint} since if $A$ is |
\item Hermitian matrices are also called \emph{self-adjoint} since if $A$ is |
| Hermitian, then in the usual |
Hermitian, then in the usual |
| inner product of $\mathbb{C}^n$, we have |
inner product of $\mathbb{C}^n$, we have |
| $$ \langle u,Av \rangle = \langle Au,v\rangle$$ |
$$ \langle u,Av \rangle = \langle Au,v\rangle$$ |
| for all $u,v\in \mathbb{C}^n$. |
for all $u,v\in \mathbb{C}^n$. |
|
|
| \end{enumerate} |
\end{enumerate} |
|
|
| \subsubsection*{Example} |
\subsubsection*{Example} |
| \begin{enumerate} |
\begin{enumerate} |
|
\item For any $n\times m$ matrix $A$, the $n\times n$ matrix $A A^\ast$ is
|
\item For any $n\times m$ matrix $A$, the $n\times n$ matrix $A A^\ast$ i |
| Hermitian. |
Hermitian. |
| \item For any square matrix $A$, the Hermitian part of $A$, |
|
| $\frac{1}{2}(A+A^\ast)$ is Hermitian. |
|
| See \PMlinkname{this page}{DirectSumOfHermitianAndSkewHermitianMatrices}. |
|
| \item |
\item |
| $$ \begin{bmatrix} |
$$ \begin{bmatrix} |
| 1 & 1 + i & 1 + 2i & 1 + 3i \\ |
1 & 1 + i & 1 + 2i & 1 + 3i \\ |
| 1 - i & 2 & 2 + 2i & 2 + 3i \\ |
1 - i & 2 & 2 + 2i & 2 + 3i \\ |
| 1 - 2i & 2 - 2i & 3 & 3 + 3i \\ |
1 - 2i & 2 - 2i & 3 & 3 + 3i \\ |
| 1 - 3i & 2 - 3i & 3 - 3i & 4 |
1 - 3i & 2 - 3i & 3 - 3i & 4 |
| \end{bmatrix} $$ |
\end{bmatrix} $$ |
| \end{enumerate} |
\end{enumerate} |
| The first two examples are also examples of normal matrices. |
|
|
|
| \subsubsection*{Notes} |
\subsubsection*{Notes} |
| Hermitian matrices are named after |
Hermitian matrices are named after |
| Charles Hermite (1822-1901) \cite{hermite}, who proved in 1855 that the |
Charles Hermite (1822-1901) \cite{hermite}, who proved in 1855 that the |
| eigenvalues of these matrices are always real \cite{eves}. |
eigenvalues of these matrices are always real \cite{eves}. |
|
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{eves} H. Eves, |
\bibitem{eves} H. Eves, |
| \emph{Elementary Matrix Theory}, |
\emph{Elementary Matrix Theory}, |
| Dover publications, 1980. |
Dover publications, 1980. |
| \bibitem{hermite} |
\bibitem{hermite} |
| The MacTutor History of Mathematics archive, |
The MacTutor History of Mathematics archive, |
| \PMlinkexternal{Charles Hermite}{http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Hermite.html} |
\PMlinkexternal{Charles Hermite}{http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Hermite.html} |
| \end{thebibliography} |
\end{thebibliography} |
