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Mathematik-Online problems:

Problem 67: Statements on Positive Definite Matrices

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
A matrix is called positive definite, if all eigenvalues are positive. Show that each symmetric, positive definite matrix $ A\in\mathbb{R}^{n\times n}$ has the following properties:
a)
For every eigenvector $ v$ of $ A$ is true: $ v^{\rm {t}}Av>0$.
b)
For every vector $ w\in\mathbb{R}^n\setminus\{0\}$ is true: $ w^{\rm {t}}Aw>0$.
c)
For each regular matrix $ T\in\mathbb{R}^{n\times n}$ is true: $ T^{\rm {\,t}}AT$ is symmetric and positive definite.

(Authors: Kimmerle/Höfert)

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[Solutions]
  automatisch erstellt am 14. 10. 2004