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TR13-103 | 24th July 2013 15:21

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 $\mathcal{B}$ of the $n \times n$ matrices over some field $\mathbb{F}$, we consider the following problems: $\textit{symbolic matrix rank}$ (SMR) is the problem to determine the maximum rank among matrices in $\mathcal{B}$, while $\textit{symbolic determinant identity testing}$ (SDIT) is the question to decide whether there exists a nonsingular matrix in $\mathcal{B}$. 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 $\mathcal{B}$ is spanned by unknown rank one matrices, answering an open question of Gurvits. Our second algorithm solves the constructive SDIT when $\mathcal{B}$ 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 over the rational numbers, and the first algorithm actually solves (the non-constructive) SMR independently from the field size. Our main tool to obtain these results is to generalize Wong sequences, a classical method to deal with pairs of matrices, to the case of pairs of matrix spaces.

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