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, 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 ${\displaystyle B}$ is the transformation matrix to the new base.

See also

• Invariant (mathematics)
• Gauge invariance
• Trace diagram