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  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 ).
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  discussed an inverse eigenvalue problem of a tridiagonal matrix under a submatrix constraint; Peng and Hu  considered an inverse eigenpair problem of a Jacobi matrix under a leading principal submatrix
constraint; Peng and Hu  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.