Augmented nodal matrices and normal trees

Augmented nodal matrices play an important role in the analysis of different features of electrical circuit models. Their study can be addressed in an abstract setting involving two- and three-colour weighted digraphs. By means of a detailed characterization of the structure of proper and normal trees, we provide a unifying framework for the rank analysis of augmented matrices. This covers in particular Maxwell’s tree-based determinantal expansions of (non-augmented) nodal matrices, which can be considered as a one-colour version of our results. Via different colour assignments to circuit devices, we tackle the DC-solvability problem and the index characterization of certain differential-algebraic models which arise in the nodal analysis of electrical circuits, extending several known results of passive circuits to the non-passive context.     

Spin models constructed from Hadamard matrices

A spin model (for link invariants) is a square matrix W which satisfies certain axioms. For a spin model W, it is known that W ^T W ^?1 is a permutation matrix, and its order is called the index of W. Jaeger and Nomura found spin models of index?2, by modifying the construction of symmetric spin models from Hadamard matrices. The aim of this paper is to give a construction of spin models of an arbitrary even index from any Hadamard matrix. In particular, we show that our spin models of indices a power of 2 are new.     

The computation and perturbation analysis for weighted group inverse of rectangular matrices

Cen (Math. Numer. Sin. 29(1):39–48, 2007) has defined a weighted group inverse of rectangular matrices. Let A∈C ^m×n ,W∈C ^n×m , if X∈C ^m×n satisfies the system of matrix equations $$(W_{1})\quad AWXWA=A,\quad\quad (W_{2})\quad XWAWX=X,\quad\quad (W_{3})\quad AWX=XWA$$ X is called the weighted group inverse of A with W, and denoted by A _W ^# . In this paper, we will study the algebra perturbation and analytical perturbation of this kind weighted group inverse A _W ^# . Under some conditions, we give a decomposition of B _W ^# ?A _W ^# . As a results, norm estimate of ‖B _W ^# ?A _W ^# ‖ is presented (where B=A+E). An expression of algebra of perturbation is also obtained. In order to compute this weighted group inverse with ease, we give a new representation of this inverse base on Gauss-elimination, then we can calculate this weighted inverse by Gauss-elimination. In the end, we use a numerical example to show our results.     

The representation and perturbation of theW-weighted Drazin inverse

LetA andE bem x n matrices andW an n xm matrix, and letA _d,W denote the W-weighted Drazin inverse ofA. In this paper, a new representation of the W-weighted Drazin inverse ofA is given. Some new properties for the W-weighted Drazin inverseA _d,W and B_d,W are investigated, whereB =A+E. In addition, the Banach-type perturbation theorem for the W-weighted Drazin inverse ofA andB are established, and the perturbation bounds for ∥B_d,W∥ and ∥B_{d, W}, -A_d,W∥/∥A_d,W∥ are also presented. WhenA andB are square matrices andW is identity matrix, some known results in the literature related to the Drazin inverse and the group inverse are directly reduced by the results in this paper as special cases.     

Equalities and inequalities for inertias of hermitian matrices with applications

The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of 2×2 block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression A-BXB^∗ with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation AX=B, as well as the extremal inertias of a partial block Hermitian matrix.     

3×3 Orthostochastic matrices and the convexity of generalized numerical ranges

Let $\text{U}$₃ be the set of all 3 × 3 unitary matrices, and let A and B be two 3 × 3 complex nor?al matrices. In this note, the authors first give a necessary and sufficient condition for a 3 × 3 doubly stochastic matrix to be orthostochastic and then use this result to consider the structure of the sets $\text{W}$ (A) = {Diag UAU¹ : U ∈ $\text{U}$₃} and W(A,B) = {Tr UAU¹B: U ∈ $\text{U}$₃}, where ¹ denotes the transpose conjugate.     

On the representability of totally unimodular matrices on bidirected graphs

We present a polynomial time algorithm to construct a bidirected graph for any totally unimodular matrix B by finding node-edge incidence matrices Q and S such that QB=S. Seymour’s famous decomposition theorem for regular matroids states that any totally unimodular (TU) matrix can be constructed through a series of composition operations called k-sums starting from network matrices and their transposes and two compact representation matrices B₁,B₂ of a certain ten element matroid. Given that B₁,B₂ are binet matrices we examine the k-sums of network and binet matrices. It is shown that thek-sum of a network and a binet matrix is a binet matrix, but binet matrices are not closed under this operation for k=2,3. A new class of matrices is introduced, the so-called tour matrices, which generalise network, binet and totally unimodular matrices. For any such matrix there exists a bidirected graph such that the columns represent a collection of closed tours in the graph. It is shown that tour matrices are closed under k-sums, as well as under pivoting and other elementary operations on their rows and columns. Given the constructive proofs of the above results regarding the k-sum operation and existing recognition algorithms for network and binet matrices, an algorithm is presented which constructs a bidirected graph for any TU matrix.     

Characterizations of optimal scalings of matrices

A scaling of a non-negative, square matrixA ≠ 0 is a matrix of the formDAD ^?1, whereD is a non-negative, non-singular, diagonal, square matrix. For a non-negative, rectangular matrixB ≠ 0 we define a scaling to be a matrixCBE ^?1 whereC andE are non-negative, non-singular, diagonal, square matrices of the corresponding dimension. (For square matrices the latter definition allows more scalings.) A measure of the goodness of a scalingX is the maximal ratio of non-zero elements ofX. We characterize the minimal value of this measure over the set of all scalings of a given matrix. This is obtained in terms of cyclic products associated with a graph corresponding to the matrix. Our analysis is based on converting the scaling problem into a linear program. We then characterize the extreme points of the polytope which occurs in the linear program.     

Multi-degree cyclic scheduling of a no-wait robotic cell with multiple robots

This paper addresses cyclic scheduling of a no-wait robotic cell with multiple robots. In contrast to many previous studies, we consider r-degree cyclic (r > 1) schedules, in which r identical parts with constant processing times enter and leave the cell in each cycle. We propose an algorithm to find the minimal number of robots for all feasible r-degree cycle times for a given r (r > 1). Consequently, the optimal r-degree cycle time for any given number of robots for this given r can be obtained with the algorithm. To develop the algorithm, we first show that if the entering times of r parts, relative to the start of a cycle, and the cycle time are fixed, minimizing the number of robots for the corresponding r-degree schedule can be transformed into an assignment problem. We then demonstrate that the cost matrix for the assignment problem changes only at some special values of the cycle time and the part entering times, and identify all special values for them. We solve our problem by enumerating all possible cost matrices for the assignment problem, which is subsequently accomplished by enumerating intervals for the cycle time and linear functions of the part entering times due to the identification of the special values. The algorithm developed is shown to be polynomial in the number of machines for a fixed r, but exponential if r is arbitrary.     

The Picard iteration and its application

The convergence behavior of the Picard iteration X_k+1=AX_k+B and the weighted case Y_k=X_k/b_k is investigated. It is shown that the convergence of both these iterations is related to the so-called effective spectrum of A with respect to some matrix. As an application of our convergence results we discuss the convergence behavior of a sequence of scaled triangular matrices {D^NT^N }.     

On solutions of matrix equations and

With the help of the Kronecker map, a complete, general and explicit solution to the Yakubovich matrix equation V−AVF=BW, with F in an arbitrary form, is proposed. The solution is neatly expressed by the controllability matrix of the matrix pair (A,B), a symmetric operator matrix and an observability matrix. Some equivalent forms of this solution are also presented. Based on these results, explicit solutions to the so-called Kalman–Yakubovich equation and Stein equation are also established. In addition, based on the proposed solution of the Yakubovich matrix equation, a complete, general and explicit solution to the so-called Yakubovich-conjugate matrix is also established by means of real representation. Several equivalent forms are also provided. One of these solutions is neatly expressed by two controllability matrices, two observability matrices and a symmetric operator matrix.     

Norms and inequalities for condition numbers,III

The main results provide comparisons between condition numbers (based on unitarily invariant norms) of (i) positive definite (Hermitian) matrices A, B and of A + B, (ii) a positive definite matrix and its principal submatrix, and (iii) a matrix and an augmented form of the matrix.     

一类无界算子矩阵的本质谱

本文研究了次对角占优的无界算子矩阵M=(ABCD)的左本质谱和本质谱.利用分析方法和分块算子的性质,得到了整个算子矩阵的本质谱(左本质谱)与其内部元素的本质谱(左本质谱)之间的关系.     

Lyapunov,Bézout,and Hankel

Given a polynomial f of degree n, we denote by C its companion matrix, and by S the truncated shift operator of order n. We consider Lyapunov-type equations of the form X?SXC=>W and X?CXS=W. We derive some properties of these equations which make it possible to characterize Bezoutian matrices as solutions of the first equation with suitable right-hand sides W (similarly for Hankel and the second equation) and to write down explicit expressions for these solutions. This yields explicit factorization formulae for polynomials in C, for the Schur-Cohn matrix, and for matrices satisfying certain intertwining relations, as well as for Bezoutian matrices.     

Spectral inclusion for unbounded block operator matrices

In this paper we establish a new analytic enclosure for the spectrum of unbounded linear operators A admitting a block operator matrix representation. For diagonally dominant and off-diagonally dominant block operator matrices, we show that the recently introduced quadratic numerical range W²(A) contains the eigenvalues of A and that the approximate point spectrum of A is contained in the closure of W²(A). This provides a new method to enclose the spectrum of unbounded block operator matrices by means of the non-convex set W²(A). Several examples illustrate that this spectral inclusion may be considerably tighter than the one by the usual numerical range or by perturbation theorems, both in the non-self-adjoint case and in the self-adjoint case. Applications to Dirac operators and to two-channel Hamiltonians are given.     

On the perturbation of weighted group inverse of rectangular matrices

Cen (Math. Numer. Sin. 29:39–48, 2007) defined a weighted group inverse of rectangular matrices. For given matrices A∈C ^m×n and W∈C ^n×m , if X∈C ^m×n satisfies $$( W_{1} )\ AWXWA=A, \qquad ( W_{2} ) \ XWAWX=X,\qquad ( W_{3} )\ AWX=XWA $$ then X is called the W-weighted group inverse, which is denoted by $A_{W}^{\#}$ . In this paper, for given rectangular matrices A and E and B=A+E, we investigate the perturbation of the weighted group inverse $A_{W}^{\#}$ and present the upper bounds for $\|B_{W}^{\#} \|$ .     

Spectral norm of products of random and deterministic matrices

We study the spectral norm of matrices W that can be factored as W?=?BA, where A is a random matrix with independent mean zero entries and B is a fixed matrix. Under the (4?+???)th moment assumption on the entries of A, we show that the spectral norm of such an m × n matrix W is bounded by ${\sqrt{m} + \sqrt{n}}$ , which is sharp. In other words, in regard to the spectral norm, products of random and deterministic matrices behave similarly to random matrices with independent entries. This result along with the previous work of Rudelson and the author implies that the smallest singular value of a random m × n matrix with i.i.d. mean zero entries and bounded (4?+???)th moment is bounded below by ${\sqrt{m} - \sqrt{n-1}}$ with high probability.     

Additive results for the Wg-Drazin inverse

In this paper we prove the formula for the expression (A+B)^d,W in terms of A,B,W,A^d,W,B^d,W, assuming some conditions for A,B and W. Here S^d,W denotes the generalized W-weighted Drazin inverse of a linear bounded operator S on a Banach space.     

Special issue: recent progress on iterative methods for large systems of equations

Let A x = b be a large and sparse system of linear equations where A is a nonsingular matrix. An approximate solution is frequently obtained by applying preconditioned iterations. Consider the matrix B = A + P Q ^T where \(P, Q \in \mathbb {R}^{n \times k}\) are full rank matrices. In this work, we study the problem of updating a previously computed preconditioner for A in order to solve the updated linear system B x = b by preconditioned iterations. In particular, we propose a method for updating a Balanced Incomplete Factorization preconditioner. The strategy is based on the computation of an approximate Inverse Sherman-Morrison decomposition for an equivalent augmented linear system. Approximation properties of the preconditioned matrix and an analysis of the computational cost of the algorithm are studied. Moreover, the results of the numerical experiments with different types of problems show that the proposed method contributes to accelerate the convergence.     

Multivariate normal distributions,Fisher information and matrix inequalities

Using appropriately parameterized families of multivariate normal distributions and basic properties of the Fisher information matrix for normal random vectors, we provide statistical proofs of the monotonicity of the matrix function A ^-1 in the class of positive definite Hermitian matrices. Similarly, we prove that A ¹¹ < A ^-111, where A ₁₁ is the principal submatrix of A and A ¹¹ is the corresponding submatrix of A ^-1. These results in turn lead to statistical proofs that the the matrix function A ^-1 is convex in the class of positive definite Hermitian matrices and that A ² is convex in the class of all Hermitian matrices. (These results are based on the Loewner ordering of Hermitian matrices, under which A < B if A - B is non-negative definite.) The proofs demonstrate that the Fisher information matrix, a fundamental concept of statistics, deserves attention from a purely mathematical point of view.