Editing Similarity invariance

{{sources|date=November 2019}}In [[linear algebra]], '''similarity invariance''' is a property exhibited by a function whose value is unchanged under similarities of its domain.  That is, <math>f</math> is invariant under similarities if <math>f(A) = f(B^{-1}AB)</math> where <math>B^{-1}AB</math> is a [[matrix (mathematics)|matrix]] [[matrix similarity|similar]] to ''A''.  Examples of such functions include the [[trace (linear algebra)|trace]], [[determinant]], [[characteristic polynomial]], and the [[minimal polynomial (linear algebra)|minimal polynomial]].

A more colloquial phrase that means the same thing as similarity invariance is "basis independence", since a matrix can be regarded as a [[linear operator]], written in a certain [[basis (linear algebra)|basis]], and the same operator in a new basis is related to one in the old basis by the conjugation <math>B^{-1}AB</math>, where <math>B</math> is the [[transformation matrix]] to the new basis.

== See also ==
* [[Invariant (mathematics)]]
* [[Gauge invariance]]
* [[Trace diagram]]

[[Category:Functions and mappings]]

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