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# Positive definite

In linear algebra, the positive definite matrices are (in several ways) analogous to the positive real numbers. An n × n Hermitian Hermitian matrix M is said to be positive definite if it has one (and therefore all) of the following 6 equivalent properties:

(1) For all non-zero vectors z in Cn we have

z* M z > 0.
Here we view z as a column vector with n complex entries and z* as the complex conjugate of its transpose.

(2) For all non-zero vectors x in Rn we have

xT M x > 0
(where xT denotes the transpose of the column vector x).

(3) For all non-zero vectors u in Zn (all components being integers), we have

uT M u > 0.

(4) All eigenvalues of M are positive.

(5) The form

<x, y> = x* M y
defines an inner product on Cn. (In fact, every inner product on Cn 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, ..., and M itself.

## 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 Rn (or, equivalently, all non-zero x in Cn). It is called positive semidefinite if

x* M x ≥ 0

for all x in Rn (or Cn) and negative semidefinite if

x* M x ≤ 0

for all x in Rn (or Cn).

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

## Generalizations

Suppose K denotes the field R or C, V is a vector space over K, and B : V × VK 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.

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