Similarity

Specific definitions

Different fields provide differing definitions of similarity:

In mathematics

In computer science

In other fields

See also

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 can be chosen to lie in H.

Properties

Similarity is an equivalence relation on the space of square matrices.

Similar matrices share any properties that are really properties of the represented linear operator: