Mathematics-Online lexicon: Annotation toEigenvalues and Eigenvectors under Similarity Transformations

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	Mathematik-Online lexicon: Annotation to
	Eigenvalues and Eigenvectors under Similarity Transformations

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

The eigenvalues remain unchanged under similarity transformation

$\displaystyle A \to B = Q^{-1}AQ\, .$

The eigenvectors are transformed according to the change of basis described by

. That is, An eigenvector

associated with eigenvalue $\lambda$ corresponds to eigenvector $v = Q^{-1}w$ of

associated with the same eigenvalue.

Let

be an eigenvector of

for the eigenvalue $\lambda$ , then $Aw=\lambda w$ . For $v=Q^{-1}w$ it follows that

$\displaystyle Bv=Q^{-1}AQv = Q^{-1}AQQ^{-1}w = Q^{-1}Aw = Q^{-1}\lambda w = \lambda Q^{-1}w = \lambda v\,.$

(Authors: Burkhardt/Höllig/Hörner)

automatisch erstellt am 9. 2. 2005