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In [[linear algebra]], thea '''positive-definite matrix''' is a [[matrixHermitian (mathematics)|matricesmatrix]] arewhich (in severalmany ways) is analogous to thea [[positive]] [[real number]]s. The notion is closely related to a [[positive-definite]] symmetric [[bilinear form]] (or a [[sesquilinear form]] in the complex case).
First, define some things:
==Equivalent formulations==
*<math>a^{T}</math> is the [[transpose]] of a matrix or vector <math>a</math>
*<math>a^{*}</math> is the [[conjugate transpose|complex conjugate of its transpose]] <math>a</math>
*<math>\mathbb{R}</math> is the set of all real numbers
*<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
AnLet ''M'' be an ''n'' × ''n'' [[Hermitian matrix]]. In the following we denote the [[transpose]] of a matrix of vector ''a'' by <math>Ma^{T}</math>, and the [[conjugate transpose]] by <math>a^{*}</math>. The matrix ''M'' 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>.
Note that the quantity <math>z^{*} M z</math> is always real.
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 \in \mathbb{R}^n</math> inwe have
<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|eigenvalueseigenvalue]]s <math>\lambda_i</math> of <math>M</math> are positive. (Recall that the eigenvalues of a Hermitian matrix are necessarily real).
:<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]] on <math>\mathbb{C}^n</math>. (In fact, every space|inner product]] on <math>\mathbb{C}^n</math> arises in this fashion from a Hermitian positive definite matrix.)
<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]]:
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