  A new necessary and sufficient condition for the existence of minor left
prime factorizations of multivariate polynomial matrices without full row rank
is presented. The key idea is to establish a relationship between a matrix and
its full row rank submatrix. Based on the new result, we propose an algorithm
for factorizing matrices and have implemented it on the computer algebra system
Maple. Two examples are given to illustrate the effectiveness of the algorithm,
and experimental data shows that the algorithm is efficient.
