The Singularity/Nonsingularity Problem for Matrices Satisfying Diagonal Dominance Conditions in Terms of Directed Graphs

The paper considers the singularity/nonsingularity problem for matrices satisfying certain conditions of diagonal dominance. The conditions considered extend the classical diagonal dominance conditions and involve the directed graph of the matrix in question. Furthermore, in the case of the so-called mixed diagonal dominance, the corresponding conditions are allowed to involve both row and column sums for an arbitrary finite set of matrices diagonally conjugated to the original matrix. Conditions sufficient for the nonsingularity of quasi-irreducible matrices strictly diagonally dominant in certain senses are established, as well as necessary and sufficient conditions of singularity/nonsingularity for weakly diagonally dominant matrices in the irreducible case. The results obtained are used to describe inclusion regions for eigenvalues of arbitrary matrices. In particular, a direct extension of the Gerschgorin (r = 1) and Ostrowski-Brauer (r = 2) theorems to r ≥ 3 is presented. Bibliography: 18 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 40–83.     

Distributions of eigenvalues and inertias of some block Hermitian matrices consisting of orthogonal projectors

A complex square matrix A is called an orthogonal projector if A ²?=?A?=?A*, where A* is the conjugate transpose of A. In this article, we first give some formulas for calculating the distributions of real eigenvalues of a linear combination of two orthogonal projectors. Then, we establish various expansion formulas for calculating the inertias, ranks and signatures of some 2?×?2 and 3?×?3, as well as k?×?k block Hermitian matrices consisting of two orthogonal projectors. Many applications of the formulas are presented in characterizing interval distributions of numbers of eigenvalues, and nonsingularity of these block Hermitian matrices. In addition, necessary and sufficient conditions are given for various equalities and inequalities of these block Hermitian matrices to hold.     

关于四个三幂等阵线性组合的可逆性和群逆

本文研究了四个三幂等阵线性组合的可逆性及群逆.利用矩阵分解的方法,获得了它们可逆及群逆的一些条件,并得到其逆和群逆的计算公式,这些结论完善了k幂等阵可逆性理论.     

Inclusion sets for the singular values of a square matrix

The paper presents a general approach to deriving inclusion sets for the singular values of a matrix A = (a_ij) ∈ ℂ ^n×n. The key to the approach is the following result: If σ is a singular value of A, then a certain matrix C(σ, A) of order 2n, whose diagonal entries are σ² − | a_ii|², i = 1, …, n, is singular. Based on this result, we use known diagonal-dominance type nonsingularity conditions to obtain inclusion sets for the singular values of A. Scaled versions of the inclusion sets, allowing one, in particular, to obtain Ky Fan type results for the singular values, are derived by passing to the conjugated matrix D⁻¹C(σ, A)D, where D is a positive-definite diagonal matrix. Bibliography: 16 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 52–77.     

On pointwise complete pairs of linear transformations

We prove the pointwise completeness of the order n system with constant coefficients under the assumption that the matrices of the system split into square blocks of the same size so that the collection of all blocks embeds into a finite dimensional associative division algebra; the block rank of the passive matrix is at most 2.     

可换对合阵组合的可逆性

先讨论两个可换对合阵P,Q线性组合aP+bQ可逆的充分必要条件及可逆时逆矩阵计算公式,再利用矩阵分解,以两种形式讨论两个可换对合阵P,Q组合aI+bP+cQ+dPQ及三个两两可换对合阵P,Q,R组合aI+bP+cQ+dPQ+eR+fPR+gQR+hPQR可逆的充分必要条件及可逆时分别给出逆矩阵计算公式.     

Filling the gap between the Gerschgorin and Brualdi theorems

The paper presents new diagonal dominance type nonsingularity conditions for n × n matrices formulated in terms of circuits of length not exceeding a fixed number r ≥ 0 and simple paths of length r in the digraph of the matrix. These conditions are intermediate between the diagonal dominance conditions in terms of all paths of length r and Brualdi’s diagonal dominance conditions, involving all the circuits. For r = 0, the new conditions reduce to the standard row diagonal dominance conditions , i = 1, ..., n, whereas for r = n they coincide with the Brualdi circuit conditions. Thus, they connect the classical Lévy-Desplanques theorem and the Brualdi theorem, yielding a family of sufficient nonsingularity conditions. Further, for irreducible matrices satisfying the new diagonal dominance conditions with nonstrict inequalities, the singularity/nonsingularity problem is solved. Also the new sufficient diagonal dominance conditions are extended to the so-called mixed conditions, simultaneously involving the deleted row and column sums of an arbitrary finite set of matrices diagonally conjugated to a given one, which, in the simplest nontrivial case, reduce to the old-known Ostrowski conditions , i = 1, ..., n, 0 ≤ α ≤ 1. The nonsingularity conditions obtained are used to provide new eigenvalue inclusion sets, depending on r, which, as r varies from 0 to n, serve as a bridge connecting the union of Gerschgorin’s disks with the Brualdi inclusion set. Bibliography: 16 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 128–148.     

On determinantal diagonal dominance conditions

The paper suggests sufficient nonsingularity conditions for matrices in terms of certain determinantal relations of diagonal dominance type, which improve and generalize some known results. These conditions are used to describe new eigenvalue inclusion sets and to derive new two-sided bounds on the determinants of matrices satisfying them. Bibliography: 8 titles.     

Some classes of nonsingular matrices and applications

Motivated by conditions that arise from results on mean first passage times matrices in Markov chains, we consider here two classes of real matrices whose elements satisfy some of these conditions, or variation thereof, and which result in the nonsingularity of their elements. The conditions are quite distinct from Ger?gorin circles-type conditions. Our results lead to a sufficient condition for matrices to have 1 as their unique positive eigenvalue.     

On nonsingularity of block two-by-two matrices

We derive necessary and sufficient conditions for guaranteeing the nonsingularity of a block two-by-two matrix by making use of the singular value decompositions and the Moore–Penrose pseudoinverses of the matrix blocks. These conditions are complete, and much weaker and simpler than those given by Decker and Keller [D.W. Decker, H.B. Keller, Multiple limit point bifurcation, J. Math. Anal. Appl. 75 (1980) 417–430], and may be more easily examined than those given by Bai [Z.-Z. Bai, Eigenvalue estimates for saddle point matrices of Hermitian and indefinite leading blocks, J. Comput. Appl. Math. 237 (2013) 295–306] from the computational viewpoint. We also derive general formulas for the rank of the block two-by-two matrix by utilizing either the unitarily compressed or the orthogonally projected sub-matrices.     

Two-frequency decomposition

Summary. In this paper, we present a preconditioner for large systems of linear equations based on the block decomposition for block-tridiagonal matrices. This decomposition is in many respects similar to the frequency-filtering method of Wittum [18] and also to the frequency-filtering decomposition of Wagner [4]–[6]. In contrast to these methods, our approach requires no pointwise filtering conditions but, as in [1], only averaged ones; this simplifies the implementation without any loss of efficiency. Theoretical analysis of the model problem leads to the convergence rate . Numerical experiments demonstrate similar convergence behaviour for a wider class of problems.Mathematics Subject Classification (2000): 65F10, 65N22     

Overlapping block diagonal dominance and existence of Liapunov functions

The main objective of this paper is to formulate a generalization of block diagonal dominance, which can be used to establish nonsingularity of matrices via overlapping diagonal blocks. A number of stability results are derived in the new setting by exploiting the well-known M-matrix properties, as well as extensions of the normalization, scaling, and alternative norm utilization. A link between generalized block diagonal dominance and vector Liapunov functions is established, which can be applied in the stability analysis of interconnected dynamic systems.     

幂等矩阵的组合的可逆性

本文研究了两个幂等矩阵P与Q的组合aP+bQ-cPQ-dQP-ePQP (其中a,b,c,d,e∈(C),a≠0,b≠0)的可逆性. 利用P-Q的可逆性及幂等矩阵的性质,得到了aP+bQ-cPQ-dQP-ePQP可逆的一些充要条件. 推广了J. J.Koliha 和 V.RakoA(c)eviA(c)[1]及Zuo Kezheng[2]的结论.     

More About Geršgorin‐type theorems

In the recent book of R.S. Varga, [3], one of two main recurring themes is that a nonsingular theorem for matrices gives rise to an equivalent eigenvalue inclusion set in the complex plane, and conversely. If such nonsingularity result can be extended via irreducibility, usually this can be used for obtaining more information about the boundary of the corresponding eigenvalue inclusion set. Here we will start with one of Geršgorin‐type theorem for eigenvalue inclusion, given in [1], (for which exists corresponding equivalent statement about nonsingularity of a particular class of matrices) and use it for proving necessary conditions for an eigenvalue to lie on the boundary of localization area. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convex sets of nonsingular and P:-Matrices

We show that the set r(A,B) (resp. c(A,B) of square matrices whose rows (resp. columns) are the independent convex combinations of the rows (resp.columns) of real matrices A and B consists entirely of nonsingular matrices if and only if BA^-1(resp. B^-1A) is a P-matrix. This imrpoves a theorem on P-Matrices proven in [2] and [3], in the context of interval nonsingularity. We also show that every real P-matrix admits a representation BA^-1 with the above property. These reseults are only partially true for complex P-matrices. Based on them we obtain a characterizaiton of complex P-matrices in terms of block partitions.     

Bounds for the Perron root, singularity/nonsingularity conditions, and eigenvalue inclusion sets

Using a unified approach based on the monotonicity property of the Perron root and its circuit extension, a series of exact two-sided bounds for the Perron root of a nonnegative matrix in terms of paths in the associated directed graph is obtained. A method for deriving the so-called mixed upper bounds is suggested. Based on the upper bounds for the Perron root, new diagonal dominance type conditions for matrices are introduced. The singularity/nonsingularity problem for matrices satisfying such conditions is analyzed, and the associated eigenvalue inclusion sets are presented. In particular, a bridge connecting Gerschgorin disks with Brualdi eigenvalue inclusion sets is found. Extensions to matrices partitioned into blocks are proposed.     

Semiconvergence of block SOR method for singular linear systems with p-cyclic matrices

In this paper, we discuss semiconvergence of the block SOR method for solving singular linear systems with p-cyclic matrices. Some sufficient conditions for the semiconvergence of the block SOR method for solving a general p-cyclic singular system are proved.     

On reduction of block matrices in a Hilbert space

We study the problem of the reduction of self-adjoint block matrices B = (B _ij ) with given graph by a group of unitary block diagonal matrices. Under the condition that the matrices B ² and B ⁴ are orthoscalar, we describe the graphs of block matrices for which this problem is a problem of *-finite, *-tame, or *-wild representation type.     

Ranks of zero patterns and sign patterns

Let F be a field with at least three elements. Zero patterns P such that all matrices over F with pattern P have the same rank are characterized. Similar results are proven for sign patterns. These results are applied to answering two open questions on conditions for formal nonsingularity of a pattern P, as well as to proving a sufficient condition on P such that all matrices over F with pattern P have the same height characteristic.