Similarity invariance explained

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, 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

B ^{-1} AB , where B is the transformation matrix to the new basis.

See also

• Invariant (mathematics)
• Gauge invariance
• Trace diagram