A Study on Rank Commutators of Special Families of Matrices.

Given two square matrices A, B of the same size, the two matrices AB and BA may have different ranks. A non-zero square matrix A is called a rankcommutator of a family L of n × n matrices if rank(AL) = rank(LA) holds for every L in L. Let L* denote the family of all rank-commutators of L. In this paper, we investigate the members of L* for the following families L of n×n matrices: all non-zero symmetric matrices; all diagonal matrices; all diagonalizable matrices. In the process, some new notions in linear algebra are created, such as "rank-symmetric matrices" and "determinant equivalent matrices", which might be useful for other study on ranks. A few problems for further study are posed at the end of the paper. [ABSTRACT FROM AUTHOR]