Consider a polynomial matrix, ˜ G$_{v}$ ( y ) such that when this polynomial is non-zero otherwise. In other words, the ij element of ˜ G$_{v}$ ( y ) is either a non-zero polynomial of order ¯ k$_{v}$, that entered into { G v ( y ) }$_{ij}$ , or zero. Next, consider all square submatrices r$_{v}$ × r$_{v}$, r$_{v}$ ≤ q$_{v}$ of ˜ G$_{v}$ ( y ) , for at least one of those determinant is non-zero; select the largest ¯ r$_{v}$ with the property that some submatrix of this dimension has a non-zero determinant, and (i) either ¯ r$_{v}$ = q$_{v}$, or (ii) determinant of any submatrix with q$_{v}$ ≥ r$_{v}$ > ¯ r$_{v}$ is zero. In case (i) define S v = I$_{q}$$_{v}$. In case (ii) construct a non-singular matrix S $^{v}$, such that for some ¯ r$_{v}$ × p full row rank matrix of polynomials, ¯ G v ( y ) , Such a matrix always exists. Then S $^{v}$Gv ( y ) has the representation where if N v ( y ) is non-zero the polynomial entries in the matrix N v ( y ) have homogeneous polynomial terms of order no less than ¯ k$_{v}$ ; and the non-zero ( q$_{v}$ − ¯ r$_{v}$ ) × p matrix G v $^{+1}$( y ) has polynomial terms only of order ≥ ¯ k$_{v}$ + 1 . 41    ¯ G $^{v}$( y ) + N v ( y ) G v $^{+1}$( y )    S v ˜ G$_{v}$ ( y ) =    ¯ G $^{v}$( y ) 0    . { ˜ G$_{v}$ ( y ) } =     { G v ( y ) }$_{ij,}$$_{¯}$ k$_{v}$ 0