# Similarity invariance

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In [linear algebra](/source/Linear_algebra), **similarity invariance** is a property exhibited by a function whose value is unchanged under similarities of its domain. That is, f {\displaystyle f} is invariant under similarities if f ( A ) = f ( B − 1 A B ) {\displaystyle f(A)=f(B^{-1}AB)} where B − 1 A B {\displaystyle B^{-1}AB} is a [matrix](/source/Matrix_(mathematics)) [similar](/source/Matrix_similarity) to *A*. Examples of such functions include the [trace](/source/Trace_(linear_algebra)), [determinant](/source/Determinant), [characteristic polynomial](/source/Characteristic_polynomial), and the [minimal polynomial](/source/Minimal_polynomial_(linear_algebra)).

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](/source/Linear_operator), written in a certain [basis](/source/Basis_(linear_algebra)), and the same operator in a new basis is related to one in the old basis by the conjugation B − 1 A B {\displaystyle B^{-1}AB} , where B {\displaystyle B} is the [transformation matrix](/source/Transformation_matrix) to the new basis.

## See also

- [Invariant (mathematics)](/source/Invariant_(mathematics))

- [Gauge invariance](/source/Gauge_invariance)

- [Trace diagram](/source/Trace_diagram)

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