Column space decomposition and partial order on matrices

Motivated by the observation that there exists one-to-one correspondence between column space decompositions and row space decompositions of a matrix, the class of matrices dominated by this matrix under '≤' is characterized in terms of characteristic of column space decompositions, where ≤ is a matrix partial order such as the star partial order, the sharp partial order, and the core partial order. The dominance property of the minus partial order over the other partial orders in the discussion resulted in providing a new definition of shorted matrix of a matrix with respect to column space decomposition. Also, extensions of a few results given in [O.M. Baksalary and G. Trenkler. Core inverse of matrices. Linear Multilinear Algebra, 58:681-697, 2010.] are presented in this paper.