Namespaces
Variants
Actions

Difference between revisions of "Similar matrices"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (links)
(→‎Comments: Similarity is an equivalence relation)
Line 5: Line 5:
 
====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.
 +
 +
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====

Revision as of 20:43, 16 October 2014


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, 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)
How to Cite This Entry:
Similar matrices. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Similar_matrices&oldid=33700
This article was adapted from an original article by T.S. Pigolkina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article