Abstract
Given a linear subspace B of the n×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.
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