Definite matrix: Difference between revisions

Revision as of 03:46, 9 August 2005

Revision as of 04:20, 19 August 2005

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Note that the quantity <math>z^{*} M z</math> is always real.

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|valign="top"| '''2.''' || ForAll all non-zero vectors[[eigenvalue]]s <math>x \inlambda_i</math> of \mathbb{R}^n<math>M</math> weare positive. (Recall that the eigenvalues of a Hermitian matrix are necessarily havereal).

:<math>\textbf{x}^{T} M \textbf{x} > 0</math>

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|valign="top"| '''3.''' || ForThe all non-zero vectors <math>u \in \mathbb{Z}^n</math> we haveform

:<math>\textbf{u}^{T} M \textbf{u} > 0</math>.

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|valign="top"| '''4.''' || All [[eigenvalue]]s <math>\lambda_i</math> of <math>M</math> are positive. (Recall that the eigenvalues of a Hermitian matrix are necessarily real).

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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 inner product on <math>\mathbb{C}^n</math> arises in this fashion from a Hermitian positive definite matrix.)

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|valign="top"| '''64.''' || All the following matrices (the leading principle minors) have a positive [[determinant]]:

* the upper left 1-by-1 corner of <math>M</math>

* the upper left 2-by-2 corner of <math>M</math>

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* <math>M</math> itself

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Analogous statements hold if ''M'' is a real [[symmetric matrix]], by replacing <math>\mathbb{C}^n</math> by <math>\mathbb{R}^n</math>, and the [[conjugate transpose]] by the [[transpose]].

== Further properties ==