Coprime fraction of 2-D rational matrices

Abstract

This paper presents a numerical method of computing a coprime fraction of a two-dimensional (2-D) rational matrix, not necessarily proper. It is achieved by searching the primary linearly dependent rows, in order from top to bottom, of the two generalized resultants. Although the procedure is an extension of the 2-D scalar and 1-D matrix cases, the extension is highly nontrivial and the ideas involved is drastically different. The result can also be used to compute greatest common divisor (GCD) of 2-D polynomial matrices without employing primitive factorizations. Since the primitive factorization does not exist in three or higher dimensional case, it may not be possible to extend the existing methods of computing GCD, which rely heavyly on the primitive factorization, to three or higher dimensional case. The procedure presented in this paper however can be so extended.