An examination is made of the minimal factorization problem for square nonsingular rational matrices with an additional constraint that the dimensions of the factors are required to be the same as the original matrix. This dimensional constraint poses certain difficulties, and in fact it is known that the first degree minimal factorization is no longer possible in this case. In other words, the prime factors (matrices which cannot be factorized) can have degrees greater than 1. The main result of this work is to present a necessary condition for prime factor of nonsingular rational matrices with disjoint sets of poles and zeros. There are indications that this condition may also be sufficient.
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