  The problem of factorization and parametrization of compactly supported
biorthogonal wavelets was reduced to that of pseudoidentity matrix pairs by
Resnikoff, Tian, and Wells in their 2001 paper. Based on a conjecture on the
pseudoidentity matrix pairs of rank 2 stated in the same paper, they proved a
theorem which gives a complete factorization result for rank 2 compactly
supported biorthogonal wavelets. In this paper, we first provide examples to
show that the conjecture is not true, then we prove a factorization theorem for
pseudoidentity matrix pairs of rank $m\ge 2$. As a consequence, our result
shows that a slightly modified version of the factorization theorem in the rank
2 case given by Resnikoff, Tian, and Wells holds. We also provide a concrete
constructive method for the rank 2 case which is determined by applying the
Euclidean algorithm to two polynomials.
