Georgia Institute of Technology College of Sciences

I will discuss a general framework for studying what can be said about the rank of a matrix A over a field K if we only know certain crude features of A. For example, what can we say about rank(A) if we only know which entries are zero and which are nonzero? Or if K = R, what if we only know the signs of the entries of A? Or K is a normed field and we only know the absolute values? Or K=C and we only know the arguments? There are many partial answers to questions like this scattered throughout the literature, and I will explain how at least some of these results can be unified through a theory of ranks of matrices over hyperfields. This is work in progress with Tianyi Zhang.