Matrix similarity

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 $B=P^{-1}AP.$ Similar matrices represent the same linear map under two (possibly) different bases, with $P$ being the change-of-basis matrix.^[1]^[2]

A transformation $A \mapsto P -1 AP$ 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

[1] 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.

[2] Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, pp. 176–178, LCCN 70097490

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