Square complex matrices A, B are said to be consimilar if A=SB S ̄-1 for some nonsingular matrix S. Consimilarity is an equivalence relation that is a natural matrix generalization of rotation of scalars in the complex plane. We survey the known forms to which a given complex matrix may be reduced by unitary consimilarity and describe a canonical form to which it may be reduced by a general consimilarity. We derive a useful criterion for two matrices to be consimilar and show that every matrix is consimilar to its own conjugate, transpose, and adjoint, to a real matrix, and to a Hermitian matrix. © 1988.
Hong, Y. P., & Horn, R. A. (1988). A canonical form for matrices under consimilarity. Linear Algebra and Its Applications, 102(C), 143–168. https://doi.org/10.1016/0024-3795(88)90324-2