Abstract :
The set of all eigenvalues (counting multiplicities) of a matrix A is denoted by σ(A), and the inertia of A is the ordered triple i(A)=(i+(A),i−(A),i0(A)), in which i+(A), i−(A) and i0(A) are the numbers of eigenvalues with positive, negative and zero real parts, respectively. An n×n sign pattern S=(sij) has sijset membership, variant{+,−,0} and the qualitative class of S is Q(S)={A=(aij)set membership, variantMn(R):sign(aij)=sij for all i,j}. The inertia of S is the set of ordered triples i(S)={i(A):Aset membership, variantQ(S)}. An n×n sign pattern S is an inertially arbitrary pattern (IAP) if (n1, n2, n3)set membership, varianti(S) for each nonnegative triple (n1, n2, n3) with n1+n2+n3=n. Consider the n×n (2r−1)-diagonal sign pattern Sn,r with positive entry (i,j) for 1less-than-or-equals, slantj−iless-than-or-equals, slantr−1 or i=j=n, negative entry (i,j) for 1less-than-or-equals, slanti−jless-than-or-equals, slantr−1 or i=j=1, and zero entry otherwise. J.H. Drew et al. proved that Sn,2 is an IAP for 2less-than-or-equals, slantnless-than-or-equals, slant7 and conjectured that Sn,2 is an IAP for ngreater-or-equal, slanted8. Gao et al. proved that Sn,n is an IAP for ngreater-or-equal, slanted2. In this paper, it is proved that (n−k, k, 0)set membership, varianti(Sn,r) for 0less-than-or-equals, slantkless-than-or-equals, slantn, 2less-than-or-equals, slantrless-than-or-equals, slantn and that Sn,n−1 is an IAP for ngreater-or-equal, slanted3.
