Editing Matrix similarity (section)

{{Short description|Equivalence under a change of basis (linear algebra)}}
{{other uses|Similarity (geometry)|Similarity transformation (disambiguation)}}
{{Distinguish|similarity matrix}}
In [[linear algebra]], two ''n''-by-''n'' [[matrix (mathematics)|matrices]] {{mvar|A}} and {{mvar|B}} are called '''similar''' if there exists an [[invertible matrix|invertible]] ''n''-by-''n'' matrix {{mvar|P}} such that
<math display="block">B = P^{-1} A P .</math>
Similar matrices represent the same [[linear map]] under two possibly different [[Basis (linear algebra)|bases]], with {{mvar|P}} being the [[change-of-basis matrix]].<ref>{{ cite book | first1 = Raymond A. | last1 = Beauregard | first2 = John B. | last2 = Fraleigh | year = 1973 | isbn = 0-395-14017-X | title = A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields | publisher = [[Houghton Mifflin Co.]] | location = Boston | url-access = registration | url = https://archive.org/details/firstcourseinlin0000beau | pages = 240–243 }}</ref><ref>{{ citation | first1 = Richard | last1 = Bronson | year = 1970 | lccn = 70097490 | title = Matrix Methods:  An Introduction | publisher = [[Academic Press]] | location = New York | pages = 176–178 }}</ref>

A transformation {{math|''A'' ↦ ''P''<sup>−1</sup>''AP''}} is called a '''similarity transformation''' or '''conjugation''' of the matrix {{mvar|A}}.  In the [[general linear group]], similarity is therefore the same as '''[[conjugacy class|conjugacy]]''', and similar matrices are also called '''conjugate'''; however, in a given [[subgroup]] {{mvar|H}} of the general linear group, the notion of conjugacy may be more restrictive than similarity, since it requires that {{mvar|P}} be chosen to lie in {{mvar|H}}.