P-matrix completions under weak symmetry assumptions

An n-by-n matrix is called a Iir-matrix if it is one of (weakly) sign-symmetric, positive, nonnegative P-matrix, (weakly) sign-symmetric, positive, nonnegative P(0,1)-matrix, or Fischer, or Koteljanskii matrix. In this paper, we are interested in Pi-matrix completion problems, that is, when a partial Pi-matrix has a Pi-matrix completion. Here, we prove that a combinatorially symmetric partial positive P-matrix has a positive P-matrix completion if the graph of its specified entries is an Pi-cycle. In general, a combinatorially symmetric partial Pi-matrix has a Pi-matrix completion if the graph of its specified entries is a 1-chordal graph. This condition is also necessary for (weakly) sign-symmetric P(0-) or P(0,1)-matrices. (C) 2000 Elsevier Science Inc. Ail rights reserved. AMS classification: 15A48.