We determine the number of n×n symmetric matrices over GF (p) that have rank r for 0 ≤ r ≤ n. In [BM2] Brent and McKay determine the number of n × n symmetric matrices over Zp that have determinant zero. Thus they determine the number of n× n symmetric matrices over Zp that have rank n. We extend their result to symmetric matrices over GF (p) and we determine the number of matrices that have ra...

The eigenproblems of the rank-one updates of the matrices have lots of applications in scientific computation and engineering such as the symmetric tridiagonal eigenproblems by the divide-andconquer method and Web search engine. Many researchers have well studied the algorithms for computing eigenvalues of Hermitian matrices updated by a rank-one matrix [1–6]. Recently, Ding and Zhou studied a ...

We examine n×nmatrices over Zm, with 0’s in the diagonal and nonzeros elsewhere. If m is a prime, then such matrices have large rank (i.e., n1/(p−1) − O(1) ). If m is a non-prime-power integer, then we show that their rank can be much smaller. For m = 6 we construct a matrix of rank exp(c √ log n log log n). We also show, that explicit constructions of such low rank matrices imply explicit cons...