positive definitePositiveDefinite2002-02-15 02:46:022006-08-05 16:29:55Definition% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\sZ}[0]{\mathbb{Z}}\subsubsection*{Introduction}
The definiteness of a matrix is an important
property that has use in many areas of mathematics and \PMlinkescapetext{even} physics.
Below are some examples:
\begin{enumerate}
\item In optimizing problems, the definiteness of the
Hessian matrix determines the quality of an extremal value.
The full details can be found on
\PMlinkname{this page}{RelationsBetweenHessianMatrixAndLocalExtrema}.
\end{enumerate}
{\bf Definition} \cite{pease}
Suppose $A$ is an $n\times n$ square Hermitian matrix.
If, for any non-zero vector $x$, we have that
$$x^\ast Ax>0,$$
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
the transpose of $x$.)
One can show that a Hermitian matrix is positive definite if
and only if all its eigenvalues are positive \cite{pease}.
Thus the determinant of a positive definite matrix
is positive, and
a positive definite matrix is always invertible.
The Cholesky decomposition provides an economical method for
solving linear equations involving a positive definite matrix.
Further conditions and properties for positive definite matrices
are given in \cite{johnson:pdm}.
\begin{thebibliography}{9}
\bibitem {pease} M. C. Pease,
\emph{Methods of Matrix Algebra},
Academic Press, 1965
\bibitem{johnson:pdm} C.R. Johnson, \emph{Positive definite matrices},
American Mathematical Monthly, Vol. 77, Issue 3 (March 1970) 259-264.
\end{thebibliography}