On some properties of polynomial matrices and rational function matrices

As we know that, in the investigation of frequency-domain theory of linear systems, column-reduced, column (row) Hermite forms of polynomial matrix, and the minimal polynomial base for null spaces of a transfer function matrix (in brief, TFM) with its defect, are basic properties. In this paper, related conclusions in current literatures on these properties are reviewed and generalized, and mistakes or deficiency in their mathematical proofs are corrected or mended.