# Similar Matrices

## 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*.## 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 ^{-1}AP.* 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*–

*A*ǀ and ǀλ

*E*–

*B*ǀ and, consequently, the determinants ǀ

*A*and ǀ

*B*ǀ and the eigenvalues of the similar matrices

*A*and

*B*coincide.