: Mathematics: similarity invariance

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similarity invariance

In linear algebra, similarity invariance is a property exhibited by a function whose value is unchanged under similarities of its domain. That is, ${\\displaystyle f}$ is invariant under similarities if ${\\displaystyle f(A)=f(B^{-1}AB)}$ where ${\\displaystyle B^{-1}AB}$ is a matrix similar to A. Examples of such functions include the trace, determinant, characteristic polynomial, 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 basis is related to one in the old basis by the conjugation ${\\displaystyle B^{-1}AB}$ , where ${\\displaystyle B}$ is the transformation matrix to the new basis.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Similarity_invariance)

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