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Square matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085110/s0851101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085110/s0851102.png" /> of the same order related by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085110/s0851103.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085110/s0851104.png" /> is a non-degenerate matrix of the same order. Similar matrices have the same rank, the same determinant, the same characteristic polynomial, and the same eigenvalues. It is often important to select a matrix similar to a given one but having a possibly simpler form, for example, a diagonal or Jordan form (see [[Jordan matrix|Jordan matrix]]).
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Square matrices $A$ and $B$ of the same order related by $B=S^{-1}AS$, where $S$ is a non-degenerate matrix of the same order. Similar matrices have the same rank, the same determinant, the same characteristic polynomial, and the same eigenvalues. It is often important to select a matrix similar to a given one but having a possibly simpler form, for example, a diagonal or Jordan form (see [[Jordan matrix|Jordan matrix]]).

Revision as of 18:17, 7 December 2012


Square matrices $A$ and $B$ of the same order related by $B=S^{-1}AS$, where $S$ is a non-degenerate matrix of the same order. Similar matrices have the same rank, the same determinant, the same characteristic polynomial, and the same eigenvalues. It is often important to select a matrix similar to a given one but having a possibly simpler form, for example, a diagonal or Jordan form (see Jordan matrix).

How to Cite This Entry:
Similar matrices. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Similar_matrices&oldid=29125
This article was adapted from an original article by T.S. Pigolkina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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