# Similarity

• C. C. Mac Duffee
Part of the Ergebnisse der Mathematik und Ihrer Grenƶgebiete book series (MATHE1, volume 5)

## Abstract

Similar matrices. Two matrices A and B with elements in a principal ideal ring V are called similar (written A = B) if there exists a unimodular matrix P such that A = P I BP.5 Similarity is an instance of equivalence, and is determinative, reflexive, symmetric and transitive (§ 22). More than this, every unimodular matrix P determines an automorphism of the ring of matrices with elements in V, for if
$$A_1 {\text{} } = {\text{} }P^I B_1 P,{\text{} }A_2 {\text{} } = {\text{} }P^I B_2 P$$
than
$$A_1 {\text{} } + {\text{} }A_2 {\text{} } = {\text{} }P^I (B_1 {\text{} } + {\text{} }B_2)P,{\text{} }A_1 A_2 {\text{} } = {\text{} }P^I (B_1 B_2)P$$
A matrix may be interpreted as a linear homogeneous transformation in vector space. From this point of view similar matrices represent the same transformation referred to different bases. All the theorems of this chapter may be interpreted from this standpoint.

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