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.