Generic canonical form of pairs of matrices with zeros Original Research Article

We consider a family of pairs of m×p and m×q matrices, in which some entries are required to be zero and the others are arbitrary, with respect to transformations (A,B)maps to(SAR1,SBR2) with nonsingular S, R1, and R2. We prove that almost all of these pairs reduce to the same pair (A0,B0) from this family, except for pairs whose arbitrary entries are zeros of a certain polynomial. The polynomial and the pair (A0,B0) are constructed by a combinatorial method based on properties of a certain graph.