View source for Similar matrices

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

====Comments====
Similar matrices arise when an [[endomorphism]] of a finite-dimensional vector space over a field (a linear map of the space to itself) is represented by matrices $A$, $B$ with respect to two different bases, the change of basis being expressed by the matrix $S$.  The rank, determinant, trace, characteristic polynomial and so forth are properties of the endomorphism.

Similarity is an [[equivalence relation]] on matrices.  Over an [[algebraically closed field]], the [[Jordan matrix]] provides a canonical representative of each similarity class.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  Paul R. Halmos, ''Finite-dimensional vector spaces'', Undergraduate texts in mathematics, Springer (1974)</TD></TR>
</table>



[[Category:Linear and multilinear algebra; matrix theory]]