Generalized Wong sequences and their applications to Edmonds' problems.

Given a linear subspace B of the n í¿ n matrices over some field F , we consider the following problems: symbolic matrix rank (SMR) asks to determine the maximum rank among matrices in B , while symbolic determinant identity testing (SDIT) asks to decide whether there exists a nonsingular matrix in B . The constructive versions of these problems ask 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. (The triangularization is not given explicitly.) Both algorithms work over fields of size ¿ n + 1 . Our framework is based on generalizing Wong sequences, a classical method to deal with pairs of matrices, to pairs of matrix spaces. Deterministic efficient algorithms for two special cases of Edmonds' problem are exhibited.A classical tool in matrix analysis called the Wong sequences is generalized.As a first result, an algorithm to compute the maximum rank in a rank-1 spanned matrix space is exhibited.The first result settles an open problem by Gurvits 18].As a second result, an algorithm to compute the maximum rank in a triangularizable matrix space is exhibited.