positive definite

Introduction

The definiteness of a matrix is an important property that has use in many areas of mathematics and even physics. Below are some examples:

1. In optimizing problems, the definiteness of the Hessian matrix determines the quality of an extremal value. The full details can be found on this page.

Definition [1] 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 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 [1]. 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 [2].

Bibliography

1

M. C. Pease, Methods of Matrix Algebra, Academic Press, 1965

2

C.R. Johnson, Positive definite matrices, American Mathematical Monthly, Vol. 77, Issue 3 (March 1970) 259-264.