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*<math>\mathbb{C}</math> is the set of all complex numbers
*<math>\mathbb{Z}</math> is the set of all [[integer]]s
*<math>M</math> is any Hermitian matrix
An ''n'' × ''n'' [[Hermitian matrix]] <math>M</math> is said to be '''positive definite''' if it has one (and therefore all) of the following six equivalent properties:
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|valign="top"| '''1.''' || For all non-zero vectors <math>z \in \mathbb{C}^n</math> we have
:<math>\textbf{z}^{*} M \textbf{z} > 0</math>.
Here we view <math>z</math> as a [[column vector]] with <math>n</math> [[complex number|complex]] entries and <math>z^{*}</math> as the [[conjugate transpose|complex conjugate of its transpose]]. (<math>z^{*} M z</math> is always real.)
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|valign="top"| '''2.''' || For all non-zero vectors <math>x</math> in
<math>\mathbb{R}^n</math> we have
:<math>\textbf{x}^{T} M \textbf{x} > 0</math>
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|valign="top"| '''3.''' || For all non-zero vectors <math>u \in \mathbb{Z}^n</math>, we have
:<math>\textbf{u}^{T} M \textbf{u} > 0</math>.
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|valign="top"| '''4.''' || All [[eigenvectors|eigenvalues]] of <math>M</math> are positive.
:<math>\lambda_i(M) > 0 \; \forall i</math>
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|valign="top"| '''5.''' || The form
:<math>\langle \textbf{x},\textbf{y}\rangle = \textbf{x}^{*} M \textbf{y}</math>
defines an [[inner product space|inner product]] on
<math>\mathbb{C}^n</math>. (In fact, every inner product on <math>\mathbb{C}^n</math> arises in this fashion from a Hermitian positive definite matrix.)
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|valign="top"| '''6.''' || All the following matrices have positive [[determinant]]:
* the upper left 1-by-1 corner of <math>M</math>
* the upper left 2-by-2 corner of <math>M</math>
* the upper left 3-by-3 corner of <math>M</math>
* ...
* <math>M</math> itself
|}
== Further properties ==
Every positive definite matrix is invertible and its inverse is also positive definite.
If <math>M</math> is positive definite and <math>r > 0</math> is a real number, then <math>r M</math> is positive definite.
If <math>M</math> and <math>N</math> are positive definite, then <math>M + N</math> is also positive definite, and if <math>M N = N M</math>, then <math>MN</math> is also positive definite.
Every positive definite matrix <math>M</math>, has at least one [[square root]] matrix <math>N</math> such that <math>N^2 = M</math>.
In fact, <math>M</math> may have infinitely many square roots, but exactly one positive definite square root.
== Negative-definite, semidefinite and indefinite matrices ==
The Hermitian matrix <math>M</math> is said to be '''negative-definite''' if
:<math>x^{*} M x < 0</math>
for all non-zero <math>x \in \mathbb{R}^n</math> (or, equivalently, all non-zero <math>x \in \mathbb{C}^n</math>). It is called '''positive-semidefinite''' if
:<math>x^{*} M x \geq 0</math>
for all <math>x \in \mathbb{R}^n</math> (or <math>\mathbb{C}^n</math>) and '''negative-semidefinite''' if
:<math>x^{*} M x \leq 0</math>
for all <math>x \in \mathbb{R}^n</math> (or <math>\mathbb{C}^n</math>).
A Hermitian matrix which is neither positive- nor negative-semidefinite is called '''indefinite'''.
== Non-Hermitian matrices ==
A real matrix ''M'' may have the property that ''x''<sup>T</sup>''Mx'' > 0 for all nonzero real vectors ''x'' without being symmetric. The matrix
:<math> \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} </math>
provides an example. In general, we have ''x''<sup>T</sup>''Mx'' > 0 for all real nonzero vectors ''x'' if and only if the symmetric part, (''M'' + ''M''<sup>T</sup>) / 2, is positive definite.
The situation for complex matrices may be different, depending on how one generalizes the inequality ''z''<sup>*</sup>''Mz'' > 0. If ''z''<sup>*</sup>''Mz'' is real for all complex vectors ''z'', then the matrix ''M'' is necessarily Hermitian. So, if we require that ''z''<sup>*</sup>''Mz'' be real and positive, then ''M'' is automatically Hermitian. On the other hand, we have that Re(''z''<sup>*</sup>''Mz'') > 0 for all complex nonzero vectors ''z'' if and only if the Hermitian part, (''M'' + ''M''<sub>*</sub>) / 2, is positive definite.
There is no agreement in the literature on the proper definition of ''positive-definite'' for non-Hermitian matrices.
== Generalizations ==
Suppose <math>K</math> denotes the [[field (mathematics)|field]] <math>\mathbb{R}</math> or <math>\mathbb{C}</math>, <math>V</math> is a [[vector space]] over <math>K</math>, and <math> : V \times V \rightarrow K</math> is a [[bilinear]] map which is Hermitian in the sense that <math>B(x, y)</math> is always the complex conjugate of <math>B(y, x)</math>.
Then <math>B</math> is called ''positive definite'' if <math>B(x, x) > 0</math> for every nonzero <math>x</math> in <math>V</math>.
== References ==
* Roger A. Horn and Charles R. Johnson. ''Matrix Analysis,'' Chapter 7. Cambridge University Press, 1985. ISBN 0-521-30586-1 (hardback), ISBN 0-521-38632-2 (paperback).
[[Category:Matrices]]
[[it:Matrice definita positiva]]
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