Difference between revisions of "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- | + | 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