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Positive-definite matrix

In linear algebra, the positive-definite matrices are (in several ways) analogous to the positive real numbers. An n × n Hermitian matrix $M$ is said to be positive definite if it has one (and therefore all) of the following six equivalent properties. First, define some things:

$a T$ is the transpose of a matrix or vector $a$
$a *$ is the complex conjugate of its transpose $a$
$\mathbb{R}$ is the set of all real numbers
$\mathbb{C}$ is the set of all complex numbers
$\mathbb{Z}$ is the set of all integers
$M$ is any Hermitian matrix

1.	For all non-zero vectors $z \in \mathbb{C}^n$ we have $\textbf{z}^{} M \textbf{z} > 0$ . Here we view $z$ as a column vector with $n$ complex entries and $z $ as the complex conjugate of its transpose. ( $z * M z$ is always real.)
2.	For all non-zero vectors $x$ in $\mathbb{R}^n$ we have $\textbf{x}^{T} M \textbf{x} > 0$
3.	For all non-zero vectors $u \in \mathbb{Z}^n$ , we have $\textbf{u}^{T} M \textbf{u} > 0$ .
4.	All eigenvalues of $M$ are positive. $\lambda_i(M) > 0 \; \forall i$
5.	The form $\langle \textbf{x},\textbf{y}\rangle = \textbf{x}^{*} M \textbf{y}$ defines an inner product on $\mathbb{C}^n$ . (In fact, every inner product on $\mathbb{C}^n$ arises in this fashion from a Hermitian positive definite matrix.)
6.	All the following matrices have positive determinant: the upper left 1-by-1 corner of $M$ the upper left 2-by-2 corner of $M$ the upper left 3-by-3 corner of $M$ ... $M$ itself

Further properties

Every positive definite matrix is invertible and its inverse is also positive definite. If $M$ is positive definite and $r > 0$ is a real number, then $r M$ is positive definite. If $M$ and $N$ are positive definite, then $M + N$ is also positive definite, and if $M N = N M$ , then $M N$ is also positive definite. Every positive definite matrix $M$ , has at least one square root matrix $N$ such that $N 2 = M$ . In fact, $M$ may have infinitely many square roots, but exactly one positive definite square root.

Negative-definite, semidefinite and indefinite matrices

The Hermitian matrix $M$ is said to be negative-definite if

x * M x < 0

for all non-zero $x \in \mathbb{R}^n$ (or, equivalently, all non-zero $x \in \mathbb{C}^n$ ). It is called positive-semidefinite if

$x^{*} M x \geq 0$

for all $x \in \mathbb{R}^n$ (or $\mathbb{C}^n$ ) and negative-semidefinite if

$x^{*} M x \leq 0$

for all $x \in \mathbb{R}^n$ (or $\mathbb{C}^n$ ).

A Hermitian matrix which is neither positive- nor negative-semidefinite is called indefinite.

Generalizations

Suppose $K$ denotes the field $\mathbb{R}$ or $\mathbb{C}$ , $V$ is a vector space over $K$ , and $: V \times V \rightarrow K$ is a bilinear map which is Hermitian in the sense that $B (x, y)$ is always the complex conjugate of $B (y, x)$ . Then $B$ is called positive definite if $B (x, x) > 0$ for every nonzero $x$ in $V$ .

Categories: Matrices

Last updated: 08-05-2005 01:56:27

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10-26-2009 08:16:03
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