On a matrix decomposition of Hartwig and Spindelböck

The main task of the paper is to demonstrate that Corollary 6 in [R.E. Hartwig, K. Spindelböck, Matrices for which A∗ and A† commute, Linear and Multilinear Algebra 14 (1984) 241–256] provides a powerful tool to investigate square matrices with complex entries. This aim is achieved, on the one hand, by obtaining several original results involving square matrices, and, on the other hand, by reestablishing some of the facts already known in the literature, often in extended and/or generalized forms. The particular attention is paid to the usefulness of the aforementioned corollary to characterize various classes of matrices and to explore matrix partial orderings.