Effective Lower Bounds on the Matrix Rank and Their Applications

Abstract

We propose an efficiently verifiable lower bound on the rank of a sparse fully indecomposable square matrix that contains two non-zero entries in each row and each column. The rank of this matrix is equal to its order or differs from it by one. Bases of a special type are constructed in the spaces of quadratic forms in a fixed number of variables. The existence of these bases allows us to substantiate a heuristic algorithm for recognizing whether a given affine subspace passes through a vertex of a multidimensional unit cube. In the worst case, the algorithm may output a computation denial warning; however, for the general subspace of sufficiently small dimension, it correctly rejects the input. The algorithm is implemented in Python. The running time of its implementation is estimated in the process of testing.