| Version 5 |
Version 4 |
| A matrix $A$ is said to be \emph{Hermitian} or \emph{self-adjoint} if |
A matrix $A$ is said to be \emph{Hermitian} or \emph{self-adjoint} if |
| $$ A = \bar{A^T} = A^* $$ |
$$ A = \bar{A^T} = A^* $$ |
| where $A^T$ is the transpose, and $\bar{A}$ is the complex conjugate. |
where $A^T$ is the transpose, and $\bar{A}$ is the complex conjugate. |
| Note that a Hermitian matrix must have real diagonal elements, as the complex conjugate of these elements must be equal to themselves. |
Note that a Hermitian matrix must have real diagonal elements, as the complex conjugate of these elements must be equal to themselves. |
| Any real symmetric matrix is Hermitian; the real symmetric matrices are a subset of the Hermitian matrices. |
Any real symmetric matrix is Hermitian; the real symmetric matrices are a subset of the Hermitian matrices. |
| An example of a Hermitian matrix is |
An example of a Hermitian matrix is |
| $$ \begin{bmatrix} |
$$ \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} $$ |
\end{bmatrix} $$ |
| Hermitian matrices are named after |
|
| Charles Hermite (1822-1901) \cite{hermite}, who proved in 1855 that the |
|
| eigenvalues of these matrices are always real \cite{eves}. |
|
| \begin{thebibliography}{9} |
|
| \bibitem {eves} H. Eves, |
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| \emph{Elementary Matrix Theory}, |
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| Dover publications, 1980. |
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| \bibitem{hermite} |
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| The MacTutor History of Mathematics archive, |
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| \PMlinkexternal{Charles Hermite}{http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Hermite.html} |
|
| \end{thebibliography} |
|
