How Much Does a Matrix of Rank k Weigh?

Matrices with very few nonzero entries cannot have large rank. On the other hand matrices without any zero entries can have rank as low as 1. These simple observations lead us to our main question. For matrices over finite fields, what is the relationship between the rank of a matrix and the number of nonzero entries in the matrix? This question motivated a summer research project collaboration among the authors (two undergraduate students and their adviser), and although the question seems natural, we were unable to find any previously published work dealing with it. We call the number of nonzero entries of a matrix A the weight of A and denote it by wt A. For matrices over finite fields, the weight of A − B is a natural way to define the distance between A and B. In coding theory the distance between vectors defined in this way is called the Hamming distance, named after Richard Hamming, a pioneer in the field of error correcting codes. The rank of A − B, denoted rk (A − B), defines a different distance between the matrices A and B. Thus wt A and rk A give two ways to measure the distance from A to the origin and our fundamental question is about the relationship between them. The background needed for this paper comes from the undergraduate courses in linear algebra, abstract algebra, and probability. We use the fundamental ideas of linear algebra over finite fields. For each prime power q we let Fq denote the unique field with q elements. There is no problem, however, in reading the rest of the paper with only the prime fields Fp (or even F2) in mind. We use the basic concepts and results of probability up through the central limit theorem. Having restricted our investigation to matrices over finite fields, we restate the fundamental question in this way: Over Fq how many m × n matrices of rank k and weight w are there? In probabilistic terms, we are asking for the distribution of the weight for matrices of rank k. We do not have the complete answer to this question, and it seems we are far from the complete answer, so there is plenty of work left to be done. The main results we offer are the average value of the weight for matrices of fixed rank and the complete description of the weight distribution for rank 1 matrices.