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- \lambda C $, where $ \lambda $ ranges over the set of eigenvalues of the matrix $A$, is studied. In the case where $ AB = CBA $, $ AC = CA $, $ BC = CB $, a canonical form for $ A, B, C$, generalizing Thompson's result for invertible $ A, B, C,$ is introduced. Also bounds for the length of pairs of matrices $ \{A, B \} $ of the form indicated are provided.