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, $f$ is invariant under similarities if $f(A) = f(B^{-1}AB)$ where $B^{-1}AB$ 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 $B^{-1}AB$, where $B$ is the transformation matrix to the new base.

See also

• Invariant (mathematics)
• Gauge invariance
• Trace diagram