Spaces of symmetric matrices containing a nonzero matrix of bounded rank

✍ Scribed by S. Friedland; R. Loewy

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 491 KB
• Volume: 287
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(98)10185-4

✦ Synopsis

Let S, (F) denote the space of all n x n symmetric matrices over the field F. Given a positive integer k such that k < n, let d(n, k, F) be the smallest integer f such that every f dimensional subspace of Sn(F) contains a nonzero matrix whose rank is at most k. It is our purpose to consider d(n,k,F) for F = N and F = C. While the computation of d(n, k, C) is quite straightforward, we point out the difficulty in evaluating d (n, k, R). We obtain partial results regarding d(n,n-2, R), and in particular show that 4 ~< d(4, 2, R) ~< 5.