P-matrix completions under weak symmetry assumptions

An n-by-n matrix is called a Iir-matrix if it is one of (weakly) sign-symme tric, positive, nonnegative P-matrix, (weakly) sign-symmetric, positive, no nnegative P-0,P-1-matrix, or Fischer, or Koteljanskii matrix. In this paper , we are interested in Pi-matrix completion problems, that is, when a parti al Pi-matrix has a Pi-matrix completion. Here, we prove that a combinatoria lly 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 combina torially symmetric partial Pi-matrix has a Pi-matrix completion if the grap h of its specified entries is a 1-chordal graph. This condition is also nec essary for (weakly) sign-symmetric P0- or P-0,P-1-matrices. (C) 2000 Elsevi er Science Inc. Ail rights reserved. AMS classification: 15A48.