Difference between revisions of "Similar matrices"
(Comment: relationship to endomorphism, cite Halmos (1974)) |
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| − | Square matrices $A$ and $B$ of the same order related by $B=S^{-1}AS$, where $S$ is a non- | + | 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==== | ====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. | 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. | ||
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| + | Similarity is an [[equivalence relation]] on matrices. Over an [[algebraically closed field]], the [[Jordan matrix]] provides a canonical representative of each similarity class. | ||
====References==== | ====References==== | ||
Latest revision as of 18:12, 7 February 2017
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 eigenvalues. 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
| [a1] | Paul R. Halmos, Finite-dimensional vector spaces, Undergraduate texts in mathematics, Springer (1974) |
Similar matrices. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Similar_matrices&oldid=33695