In 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} 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 polyonmial , 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 \( {\displaystyle B^{-1}AB} \), where B {\displaystyle B} B is the transformation matrix to the new base.
See also
Invariant (mathematics)
Gauge invariance
Trace diagram
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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