The commutation matrix was first introduced in statistics as a transposition matrix by Murnaghan in 1938. In this paper, we first investigate the commutation matrix which is employed to transform a matrix into its transpose. We then extend the concept of the commutation matrix to commutation tensor and use the commutation tensor to achieve the unification of the two formulae of the linear preserver of the matrix rank, a classical result of Marcus in 1971.

This is continuation of the recent work by Xu, Xu and Zhang [Revisiting Hua–Marcus–Bellman–Ando inequalities on contractive matrices, Linear Algebra Appl. 430 (2009), pp. 1499–1508] on contractive matrices. We study the relations of block matrices of Hua type, present some properties that the eigenvalues of Hua matrices possess, especially for the 2 × 2 block case, discuss the analogues for higher dimensions and estimate the closeness of two Hua matrices. At the end, we propose a conjecture on the eigenvalues of Hua matrices and an open problem on the symmetric functions of the eigenvalues of contractive matrices.