| Version current |
Version 6 |
| \subsubsection*{Introduction} |
\subsubsection*{Introduction} |
| The definiteness of a matrix is an important |
The definiteness of a matrix is an important |
|
property that has use in many areas of mathematics and \PMlinkescapetext{even} physics.
|
property that has use in many areas of mathematics and even physics.
|
| Below are some examples: |
Below are some examples: |
|
|
| \begin{enumerate} |
\begin{enumerate} |
| \item In optimizing problems, the definiteness of the |
\item In optimizing problems, the definiteness of the |
| Hessian matrix determines the quality of an extremal value. |
Hessian matrix determines the quality of an extremal value. |
| The full details can be found on |
The full details can be found on |
| \PMlinkname{this page}{RelationsBetweenHessianMatrixAndLocalExtrema}. |
\PMlinkname{this page}{RelationsBetweenHessianMatrixAndLocalExtrema}. |
| \end{enumerate} |
\end{enumerate} |
|
|
|
|
| {\bf Definition} \cite{pease} |
{\bf Definition} \cite{pease} |
| Suppose $A$ is an $n\times n$ square Hermitian matrix. |
Suppose $A$ is an $n\times n$ square Hermitian matrix. |
| If, for any non-zero vector $x$, we have that |
If, for any non-zero vector $x$, we have that |
| $$x^\ast Ax>0,$$ |
$$x^\ast Ax>0,$$ |
| then $A$ a \emph{positive definite} matrix. (Here $x^\ast=\overline{x}^t$, |
then $A$ a \emph{positive definite} matrix. (Here $x^\ast=\overline{x}^t$, |
| where $\overline{x}$ is the complex conjugate of $x$, and $x^t$ is |
where $\overline{x}$ is the complex conjugate of $x$, and $x^t$ is |
| the transpose of $x$.) |
the transpose of $x$.) |
|
|
| One can show that a Hermitian matrix is positive definite if |
One can show that a Hermitian matrix is positive definite if |
| and only if all its eigenvalues are positive \cite{pease}. |
and only if all its eigenvalues are positive \cite{pease}. |
| Thus the determinant of a positive definite matrix |
Thus the determinant of a positive definite matrix |
| is positive, and |
is positive, and |
| a positive definite matrix is always invertible. |
a positive definite matrix is always invertible. |
| The Cholesky decomposition provides an economical method for |
The Cholesky decomposition provides an economical method for |
| solving linear equations involving a positive definite matrix. |
solving linear equations involving a positive definite matrix. |
| Further conditions and properties for positive definite matrices |
Further conditions and properties for positive definite matrices |
| are given in \cite{johnson:pdm}. |
are given in \cite{johnson:pdm}. |
| |
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem {pease} M. C. Pease, |
\bibitem {pease} M. C. Pease, |
| \emph{Methods of Matrix Algebra}, |
\emph{Methods of Matrix Algebra}, |
| Academic Press, 1965 |
Academic Press, 1965 |
| \bibitem{johnson:pdm} C.R. Johnson, \emph{Positive definite matrices}, |
\bibitem{johnson:pdm} C.R. Johnson, \emph{Positive definite matrices}, |
| American Mathematical Monthly, Vol. 77, Issue 3 (March 1970) 259-264. |
American Mathematical Monthly, Vol. 77, Issue 3 (March 1970) 259-264. |
| \end{thebibliography} |
\end{thebibliography} |
