More generally, the number of principal minors that must be checked is reduced when a definite

principal submatrix can be identified.

Remember that a matrix M is called P-matrix if the determinant of every

principal submatrix of M is positive (see Murty [9] and Cottle et al.

Similarly, let [A.sup.[alpha].sub.[alpha]] be a

principal submatrix of A whose rows and columns are indexed by [alpha].

The

principal submatrix [H.sub.k] of H in (3.2) can be factored in a similar way with a companion matrix revealing the eigenvalues of [H.sub.k], which are the ordinary Ritz values.

Then [A.sup.[alpha].sub.[alpha]] denotes the

principal submatrix determined by the rows and columns indexed by [alpha].

where the coefficient matrix (I - [alpha][[??].sub.11]) is the nontrivial leading

principal submatrix of (I - [alpha]H) and it is nonsingular (Theorem 6.(4.16) of [23]).

Noticing that [A.sub.kk] is a

principal submatrix of A, we call the matrix [[bar.A].sub.kk] a principal quasi-submatrix corresponding to the cluster [G.sup.k], k = 1, ..., d.

Given [lambda] [??] n, first notice that any matrix in [g.sub.n,[lambda]]([F.sub.q]) has a leading

principal submatrix of type [mu] where [mu] [??] [lambda].

Zhang, Least squares solutions to AX = B for bisymmetric matrices under a central

principal submatrix constraint and the optimal approximation, Numer.

If v [member of] V(G), let [L.sub.v](G) be the

principal submatrix of L(G) obtained by deleting the row and column corresponding to the vertex v.

For example, Deift and Nanda [4] discussed an inverse eigenvalue problem of a tridiagonal matrix under a submatrix constraint; Peng and Hu [16] considered an inverse eigenpair problem of a Jacobi matrix under a leading

principal submatrix constraint; Peng and Hu [17] studied a inverse problem of bi-symmetric matrices with a leading

principal submatrix constraint, for more we refer the reader to [6, 12, 24].

where [H.sub.k] is the

principal submatrix of order k of H, [H.sub.k] is the same matrix appended with the k first entries of the (k + 1)st row of H, and [e.sub.k] is the last column of the identity matrix of order k.