Generalized Wong Sequences and Their Applications to Edmonds’ Problems

We design two deterministic polynomial-time algorithms for variants of a problem introduced by Edmonds in 1967: determine the rank of a matrix M whose entries are homogeneous linear polynomials over the integers. Given a linear subspace B of the n × n matrices over some field F, we consider the following problems: symbolic matrix rank (SMR) is the problem to determine the maximum rank among matrices in B, while symbolic determinant identity testing (SDIT) is the question to decide whether there exists a nonsingular matrix in B. The constructive versions of these problems are asking to find a matrix of maximum rank, respectively a nonsingular matrix, if there exists one. Our first algorithm solves the constructive SMR when B is spanned by unknown rank one matrices, answering an open question of Gurvits. Our second algorithm solves the constructive SDIT when B is spanned by triangularizable matrices, but the triangularization is not given explicitly. Both algorithms work over fields of size at least n+ 1, and the first algorithm actually solves (the non-constructive) SMR independent of the field size. Our framework is based on a generalization of Wong sequences, a classical method to deal with pairs of matrices, to the case of pairs of matrix spaces