Singular spaces of matrices and their application in combinatorics

We study linear spaces of n × n matrices in which every matrix is singular. Examples are given to illustrate that a characterization of such subspaces would solve various open problems in combinatorics and in computational algebra. Several important special cases of the problem are solved, although often in disguise. 1. The problem Let A be a linear subspace of the space IR n×n of real n × n matrices. We say that A is singular if every matrix in A is singular. We are interested in the problem of characterizing singular spaces of matrices, and in obtaining an efficient algorithm to determine if a space of matrices (given by a linear basis) is singular. Unfortunately, we cannot give a complete solution to these problems, but special cases with combinatorial applications will be solved. Geometrically, det X = 0 defines a surface in IR n×n and we are interested in the linear subspaces contained in this surface. Clearly, we may restrict our attention to the maximal singular subspaces. The problem arose in differential geometry (see Room (1938)), but my interest in this problem stems from its connection to matching problems and other fundamental problems in combinatorics. In this context, the problem was formulated by Edmonds (1967), who pointed out its relevance to combinatorial algorithms and to the theory of computational complexity. We may slightly generalize the problem by considering a linear subspace A of real n × m matrices. We define the generic rank gr(A) of such a subspace as the maximum rank of matrices in it, and want to find an efficient way to compute this generic rank, given (say) a basis for A. (Unfortunately, no really efficient algorithm is known for this problem, at least if we do not allow randomization.) However, it is easy to reduce this seemingly more general problem to the problem of characterizing singular matrix spaces. A trivial upper bound on the generic rank is the column range rank lrr(A) of the matrix space, defined as the dimension of the subspace spanned by all the columns of all the matrices in A. This number is easy to compute if we have a 1