Geodesic
Commutativity of matrices up to a matrix factor
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXII, Tome 482 (2019), pp. 151-168

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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. 
@article{ZNSL_2019_482_a10,
     author = {N. A. Kolegov and O. V. Markova},
     title = {Commutativity of matrices up to a matrix factor},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {151--168},
     publisher = {mathdoc},
     volume = {482},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2019_482_a10/}
}
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N. A. Kolegov; O. V. Markova. Commutativity of matrices up to a matrix factor. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXII, Tome 482 (2019), pp. 151-168. http://geodesic.mathdoc.fr/item/ZNSL_2019_482_a10/