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 λE – Aǀ and ǀλE – Bǀ and, consequently, the determinants ǀA and ǀBǀ and the eigenvalues of the similar matrices A and B coincide.