Equivalence under a change of basis (linear algebra)
For other uses, see Similarity (geometry) and Similarity transformation (disambiguation).
Not to be confused with similarity matrix.
In linear algebra, two n-by-n matrices A and B are called similar if there exists an invertible n-by-n matrix P such that
Similar matrices represent the same linear map under two (possibly) different bases, with P being the change of basis matrix.[1][2]
A transformation A ↦ P−1AP 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.
^Beauregard, Raymond A.; Fraleigh, John B. (1973). A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields. Boston: Houghton Mifflin Co. pp. 240–243. ISBN 0-395-14017-X.
^Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, pp. 176–178, LCCN 70097490
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