The (weakly) sign symmetric P-matrix completion problems

Abstract

Abstract. In this paper it is shown that a partial sign symmetric P -matrix, whose digraph of specified entries is a symmetric n-cycle with n ≥ 6, can be completed to a sign symmetric Pmatrix. The analogous completion property is also established for a partial weakly sign symmetric P -matrix and for a partial weakly sign symmetric P 0 -matrix. Patterns of entries for 4 × 4 matrices are classified as to whether or not a partial (weakly) sign symmetric P -or weakly sign symmetric P 0 -matrix specifying the pattern must have completion to the same type of matrix. The relationship between the weakly sign symmetric P -and sign symmetric P -matrix completion problems is also examined.

The matrix completion problem regarding various classes of P0,1- matrices

“…DeAlba et al proved the symmetric 6-cycle has (weakly) sign symmetric P-completion, as does the symmetric n-cycle for n > 6 [5]. They also proved that the asymmetric n-cycle has (weakly) sign symmetric P-completion for all n > 4 and classified all order 4 digraphs as to completion for these classes.…”

“…4 DetA and 3x3 principal minors of A in Case 3B Table 4. 5 DetA and 3x3 principal minors of A in Case 3C Table 4. 6 DetA and 3x3 principal minors of A in Case 3D Table 4.7…”

“…Therefore, it also does not have weakly sign symmetric Pa. 1 -completion by Corollary 2.3. • q -6, n = 4 q = 4, n = 16 q = b, n = The digraphs q -6, n = 4-8 and q = 7, n = matrix satisfying the pattern with digraph q = 7. n -2; g -4. n = 16; g = 5, n = 7; g = 6, n = 4, 7 bos weakly sign symmetric Po^-completion.Proof: We may complete these matrices in the same manner as in Lemma 4.2 in[5].1 &12 Xi3 Oi4Consider the digraph q = 7, n = 2 shown inFigure 2.1. weakly sign symmetric PQ. ,-matrix specifying the digraph q = 7, n = 2.…”

“…Digraphs that do not have sign symmetric Pq,i-completion Proof: Any partial sign symmetric P-matrix specifying either of the order two digraphs Ç = 0; q = 1 or any of the order three digraphs q = 0; q = 1; q = 2, n = 2, 3,4; q -3, n -3 may be completed to a sign symmetric P-matrix[5]. Since these digraphs are also asymmetric,they have sign symmetric Po,i-completion by Theorem 4.1.…”

“…The following digraphs have sign symmetric Po.i-completion by Theorem 4.5 because each block has sign symmetric Po ; i-matrix completion by Lemma 4.6: g = 0; g = 1; g = 2, n = 1-5; g = 3, « = 1-11, 13; g = 4, n =1-12, 21-23, 25-27; g = 5, n =1-5, 26-28; g = 6, n =1-3; g = 7, n = 4-5; g = 8, n = 1; g = 12.Each of the digraphs q = 4, n = 16-19; q = 5, n = 29,31,33-34, 36-37; q = 6, n = 46 are asymmetric and have sign symmetric P-completion[5]. Therefore, any partial sign symmetric Po : i-matrix specifying one of these digraphs has sign symmetric Po.…”