Matrix similarity

In linear algebra, two n-by-n matrices A and B are called similar if

for some invertible n-by-n matrix P. Similar matrices represent the same linear operator under two different bases, with P being the change of basis matrix.

A transformation is called a similarity transformation or conjugation of the matrix A. In the general linear group, similarity is therefore the same as conjugacy, and similar matrices are also called conjugate; however in a given subgroup H of the general linear group, the notion of conjugacy may be more restrictive than similarity, since it requires that P be chosen to lie in H.