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1.
We show that if A is an M-matrix for which the length of the longest simple cycle in its associated undirected graph G(A) is at most 3, then every minor of A has determined sign (nonnegative or nonpositive), independent of the magnitudes of the matrix entries. Consequently, if A and B are M-matrices such that G(A) and G(B) are subgraphs of an undirected graph with longest simple cycle at most 3, then all principal minors of AB are nonnegative. 相似文献
2.
A real square matrix A is called a P-matrix if all its principal minors are positive. Such a matrix can be characterized by the sign non-reversal property. Taking a cue from this, the notion of a P-operator is extended to infinite dimensional spaces as the first objective. Relationships between invertibility of some subsets of intervals of operators and certain P-operators are then established. These generalize the corresponding results in the matrix case. The inheritance of the property of a P-operator by the Schur complement and the principal pivot transform is also proved. If A is an invertible M-matrix, then there is a positive vector whose image under A is also positive. As the second goal, this and another result on intervals of M-matrices are generalized to operators over Banach spaces. Towards the third objective, the concept of a Q-operator is proposed, generalizing the well known Q-matrix property. An important result, which establishes connections between Q-operators and invertible M-operators, is proved for Hilbert space operators. 相似文献
3.
R.J. Plemmons 《Linear algebra and its applications》1977,18(2):175-188
The purpose of this survey is to classify systematically a widely ranging list of characterizations of nonsingular M-matrices from the economics and mathematics literatures. These characterizations are grouped together in terms of their relations to the properties of (1) positivity of principal minors, (2) inverse-positivity and splittings, (3) stability and (4) semipositivity and diagonal dominance. A list of forty equivalent conditions is given for a square matrix A with nonpositive off-diagonal entries to be a nonsingular M-matrix. These conditions are grouped into classes in order to identify those that are equivalent for arbitrary real matrices A. In addition, other remarks relating nonsingular M-matrices to certain complex matrices are made, and the recent literature on these general topics is surveyed. 相似文献
4.
Charles R. Johnson D. D. Olesky P. van den Driessche 《Linear and Multilinear Algebra》1984,16(1):29-38
Sufficient conditions are given for powers and products of M-matrices to have all principal minors positive. Several of these conditions involve directed graphs of the matrices. In particular we show that if A and B are irreducible M-matrices which have longest simple circuit of length two with A+B having no simple circuit longer than three, then the product AB has all principal minors positive. 相似文献
5.
Ky Fan defines an N-matrix to be a matrix of the form A = tI ? B, B ? 0, λ < t < ?(B), where ?(B) is the spectral radius of B and λ is the maximum of the spectral radii of all principal submatrices of B of order n ? 1. In this paper, we define the closure (N0-matrices) of N-matrices by letting λ ? t. It is shown that if A ∈ Z and A-1 < 0, then A ∈ N. Certain inequalities of N-matrices are shown to hold for N0-matrices, and a method for constructing an N-matrix from an M-matrix is given. 相似文献
6.
Let A, B be n × n matrices with entries in a field F. We say A and B satisfy property if B or Bt is diagonally similar to A. It is clear that if A and B satisfy property , then they have equal corresponding principal minors, of all orders. The question is to what extent the converse is true. There are examples which show the converse is not always true. We modify the problem slightly and give conditions on a matrix A which guarantee that if B is any matrix which has the same principal minors as A, then A and B will satisfy property . These conditions on A are formulated in terms of ranks of certain submatrices of A and the concept of irreducibility. 相似文献
7.
8.
The concepts of matrix monotonicity, generalized inverse-positivity and splittings are investigated and are used to characterize the class of all M-matrices A, extending the well-known property that A?1?0 whenever A is nonsingular. These conditions are grouped into classes in order to identify those that are equivalent for arbitrary real matrices A. It is shown how the nonnegativity of a generalized left inverse of A plays a fundamental role in such characterizations, thereby extending recent work by one of the authors, by Meyer and Stadelmaier and by Rothblum. In addition, new characterizations are provided for the class of M-matrices with “property c”; that is, matrices A having a representation A=sI?B, s>0, B?0, where the powers of converge. Applications of these results to the study of iterative methods for solving arbitrary systems of linear equations are given elsewhere. 相似文献
9.
We consider intervals of matrices with respect to the usual entrywise partial ordering. Necessary and sufficient conditions are given for an interval of matrices to contain only P-matrices (i.e. matrices having only positive principal minors) or related matrices. 相似文献
10.
Wallace C. Pye 《Mathematical Programming》1992,57(1-3):439-444
This paper demonstrates that within the class of thosen × n real matrices, each of which has a negative determinant, nonnegative proper principal minors and inverse with at least one positive entry, the class ofQ-matrices coincides with the class of regular matrices. Each of these classes of matrices plays an important role in the theory of the linear complementarity problem. Lastly, analogous results are obtained for nonsingular matrices which possess only nonpositive principal minors. 相似文献
11.
Juan Manuel Pe?a 《Advances in Computational Mathematics》2011,35(2-4):357-373
We analyze a class of matrices generalizing strictly diagonally dominant matrices and included in the important class of H-matrices. Adequate pivoting strategies and the corresponding Schur complements are studied. A new class of matrices with all their principal minors positive is presented. 相似文献
12.
Let be field, and let A and B be n × n matrices with elements in . Suppose that A is completely reducible and that B is symmetric. If the principal minors of A determined by the 1- and 2-circuits of the graph of B and by the chordless circuits of the graph of A are equal to the corresponding principal minors of B, then A is diagonally similar to B; and conversely. 相似文献
13.
An n-by-n real matrix A enjoys the “leading implies all” (LIA) property, if, whenever D is a diagonal matrix such that A+D has positive leading principal minors (PMs), all PMs of A are positive. Symmetric and Z-matrices are known to have this property. We give a new class of matrices (“mixed matrices”) that both unifies and generalizes these two classes and their special diagonal equivalences by also having the LIA property. “Nested implies all” (NIA) is also enjoyed by this new class. 相似文献
14.
A real n × n matrix M is a Q-matrix if the linear complementarity problem w ? Mz=q, w ? 0, z ? 0, wtz=0 has a solution for all real n-vectors q. M is nondegenerate if all its principal minors are nonzero. Spherical geometry is applied to the problem of characterizing nondegenerate Q-matrices. The stability of 3 × 3 nondegenerate Q-matrices and a generalization of the partitioning property of P-matrices are rather easily proved using spherical geometry. It is also proved that the set of 4 × 4 nondegenerate Q-matrices is not open. 相似文献
15.
Square matrices with positive leading principal minors, called WHS-matrices (weak Hawkins–Simon), are considered in economics. Some sufficient conditions for a matrix to be a WHS-matrix after suitable row and/or column permutations have recently appeared in the literature. New and unified proofs and generalizations of some results to rectangular matrices are given. In particular, it is shown that if left multiplication of a rectangular matrix A by some nonnegative matrix is upper triangular with positive diagonal, then some row pemutation of A is a WHS-matrix. For a nonsingular A with either the first nonzero entry of each of its rows positive or the last nonzero entry of each column of A ?1 positive, again some row permutation of A is a WHS-matrix. In addition, any rectangular full rank semipositive matrix is shown to be permutation equivalent to a WHS-matrix. 相似文献
16.
Generalizations of M-matrices are studied, including the new class of GM-matrices. The matrices studied are of the form sI-B with B having the Perron-Frobenius property, but not necessarily being nonnegative. Results for these classes of matrices are shown, which are analogous to those known for M-matrices. Also, various splittings of a GM-matrix are studied along with conditions for their convergence. 相似文献
17.
A right R-module M is called simple-direct-injective if, whenever, A and B are simple submodules of M with A?B, and B?⊕M, then A?⊕M. Dually, M is called simple-direct-projective if, whenever, A and B are submodules of M with M∕A?B?⊕M and B simple, then A?⊕M. In this paper, we continue our investigation of these classes of modules strengthening many of the established results on the subject. For example, we show that a ring R is uniserial (artinian serial) with J2(R) = 0 iff every simple-direct-projective right R-module is an SSP-module (SIP-module) iff every simple-direct-injective right R-module is an SIP-module (SSP-module). 相似文献
18.
Patrick K. Torres 《Linear and Multilinear Algebra》2018,66(4):769-775
Invertibility of all convex combinations of A and I is equivalent to the real eigenvalues of A, if any, being positive. Invertibility of all matrices whose rows are convex combinations of the respective rows of A and I is equivalent to A having positive principal minors (i.e. being a P-matrix). These results are extended by considering convex combinations of higher powers of A and of their rows. The invertibility of matrices in these convex hulls is associated with the eigenvalues of A lying in open sectors of the right-half plane and provides a general context for the theory of matrices with P-matrix powers. 相似文献
19.
Jong-Shi Pang 《Mathematical Programming》1981,20(1):348-352
This note presents a class ofQ-matrices which includes Saigal's classN ofQ-matrices with negative principal minors and the classE of strictly semi-monotoneQ-matrices.Research supported by the Office of Naval Research under Contract N00014-75-C-0621 NR 047-048. 相似文献
20.
Yuri A. Kordyukov 《Mathematische Nachrichten》2002,245(1):104-128
We consider a (hypo)elliptic pseudodifferential operator Ah on a closed foliated manifold (M,ℱ), depending on a parameterh > 0, of the form Ah = A+hmB, where A is a formally self–adjoint tangentially elliptic operator of orderμ > 0 with the nonnegative principal symbol and B is a formally self–adjoint classical pseudodi.erential operator of orderm > 0 on M with the holonomy invariant transversal principal symbol such that its principal symbol is positive, if μ < m, and its transversal principal symbol is positive, if μ ≥ m. We prove an asymptotic formula for the eigenvalue distribution function Nh(λ) of the operator Ah when h tends to 0 and λ is constant. 相似文献