Fundamental factorization theorems for rational matrices over complex or real fields

It is shown that any complex rational matrix G(s) can be factorized over the complex field as a minimal product of n complex rational matrices, each of which has McMillan degree 1, where n is the McMillan degree of G(s). When confined to the real field, G(s) with real coefficient can be factorized as a minimal product of a sequence of real rational matrices with McMillan degree 1 or 2, where the McMillan degrees of the factors add up to n. The results obtained thus settle a long-standing conjecture over the factorization of rational matrices. The proofs of these fundamental factorization theorems are constructive. Therefore, by following the suggested numerical procedures, all the factors over the real or the complex fields can be easily obtained. It is noted that the results obtained can be applied in the design of multiport or multichannel filters which can be realized in a VLSI implementation.<>