Univalent function: Difference between revisions
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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 [[similar (linear algebra)|similar]] to ''A''. Examples of such functions include the [[trace (matrix)|trace]], [[determinant]], 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, and the same operator in a new base is related to one in the old base by the conjugation <math> B^{-1}AB </math>, where <math> B </math> is the [[transformation matrix]] to the new base. | |||
== See also == | |||
* [[Invariant (mathematics)]] | |||
* [[Gauge invariance]] | |||
* [[Trace diagram]] | |||
[[Category:Functions and mappings]] | |||
{{mathanalysis-stub}} | |||
Revision as of 06:48, 9 January 2014
In linear algebra, similarity invariance is a property exhibited by a function whose value is unchanged under similarities of its domain. That is, is invariant under similarities if where is a matrix similar to A. Examples of such functions include the trace, determinant, and the 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, and the same operator in a new base is related to one in the old base by the conjugation , where is the transformation matrix to the new base.
