Commutativity of Matrices Up to a Matrix Factor

The matrix relation AB = CBA is investigated. An explicit description of the space of matrices B satisfying this relation is obtained for an arbitrary fixed matrix C and a diagonalizable matrix A. The connection between this space and the family of right annihilators of the matrices A−λC, where λ ranges over the set of eigenvalues of the matrix A, is studied. In the case where AB = CBA, AC = CA, and BC = CB, a canonical form for A,B,C, generalizing Thompson’s result for invertible A,B,C, is introduced. Also bounds for the lengths of pairs of matrices {A,B} of the form indicated are provided. Bibliography: 26 titles.