# Similar Matrices

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## similar matrices

[¦sim·i·lər ′mā·tri‚sēz]
(mathematics)
Two square matrices A and B related by the transformation B = SAT, where S and T are nonsingular matrices and T is the inverse matrix of S.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Similar Matrices

Two square matrices A and B of order n are said to be similar if there exists a nonsingular, or invertible, matrix P of order n such that B= P-1AP. Similar matrices are obtained when the matrix of a linear transformation is given in different coordinate systems. The role of the matrix P in this case is played by the matrix of the transformation of coordinates. For a given matrix A it is often important to select a second matrix B that is similar to A and has as simple a form as possible—for example, the Jordan matrix. Similar matrices are of identical rank. The characteristic polynomials ǀ λEAǀ and ǀλEBǀ and, consequently, the determinants ǀA and ǀBǀ and the eigenvalues of the similar matrices A and B coincide.

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