Definite matrix: Difference between revisions

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The '''positive definite''' [[matrix|matrices]] are in several senses analogous to the positive [[real number]]s. An ''n'' × ''n'' [[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 '''C'''^''n'' we have

:<i>xz</i>^* <i>M xz</i> > 0.

Here we view ''z'' as a column vector with ''n'' [[complex number|complex]] entries and ''z''^* as the complex

for all non-zero vectors ''x'' in

'''R'''^''n'' (or, equivalently, for all non-zero ''x'' in '''C'''^''n''). Here, we view ''x'' as a column vector and ''x''^* as the complex

conjugate of its transpose.

for'''(2)''' For all non-zero vectors ''x'' in

Equivalently, a matrix ''M'' is positive definite if

'''R'''^''n'' we have

:''x''^T ''M x'' > 0

(where ''x''^T denotes the transpose of the column vector ''x'').

:<''x'(3)','' For all non-zero vectors ''yu''> =in ''x'Z'''^*''n'' ''M''(all ''y''components being [[integer]]s), we have

:''u''^T ''M u'' > 0.

'''(4)''' All [[eigenvectors|eigenvalues]] of ''M'' are positive.

'''(5)''' The form

:<''x'', ''y''> = ''x''^* ''M'' ''y''

defines an [[inner product space|inner product]] on

'''RC'''^''n''. (or,In equivalentlyfact, every inner product on '''C'''^''n''). arises in this fashion from a Hermitian 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.

== 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 ''rM'' is positive definite. If ''M'' and ''N'' are positive definite, then ''M + N'' is also positive definite, and if ''MN'' = ''NM'', then ''MN'' is also positive definite. To every positive definite matrix ''M'', there exists precisely one [[square root]]: a positive definite matrix ''N'' with ''N''² = ''M''.

== Negative definite, semidefinite and indefinite matrices ==

The Hermitian matrix ''M'' is said to be '''negative definite''' if