7.05.2014

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 its weakening, 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 finite 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.