Matrix Pair Research Articles

A note on simple nonzero finite generalized singular values

In this paper, we study the sensitivity and second order perturbation expansions of simple nonzero finite generalized singular values of a complex matrix pair, which is analytically dependent on several parameters. Our results generalize the perturbation analysis given by Sun (1988) [20] for simple nonzero singular values.

Solvatochromism, multiphoton fluorescence, and resonance energy transfer in a new octupolar dye-pair

A new octupolar molecule (E,E,E)-2,4,6-Tris [2-(4-N, N-diphenylaminophenyl) vinyl] pyridine (DPATSP) has been synthesized and studied by steady state and time-resolved fluorescence in condensed phase. The large π-conjugation along the chain has been found to be responsible for large solvent-induced shift of fluorescence. Pumping with fs pulses at 790nm have been found to induce the two-photon fluorescence. The two-photon absorption (2PA) cross section has been measured by the two photon induced fluorescence (2PIF) method. The N-methyl derivative of DPATSP shows a red shift in the absorption spectrum. The overlap of the fluorescence spectrum of DPATSP with the absorption spectrum of the DPATSP-Me makes a Förster resonance energy transfer (FRET) system. The FRET efficiency has been studied by embedding this dye pair in a polymer matrix.

Notes on Greub-Rheinboldt inequalities

Abstract In this paper, we focus on matrix Greub-Rheinboldt inequalities for commutative positive definite Hermitian matrix pairs. Some improvements, which yield sharpened bounds compared with existing results, are presented.

Fast Reduction of Generalized Companion Matrix Pairs for Barycentric Lagrange Interpolants

For a barycentric Lagrange interpolant $p(z)$, the roots of $p(z)$ are exactly the eigenvalues of a generalized companion matrix pair $(\mathbf{A},\mathbf{B})$. For real interpolation nodes, the matrix pair $(\mathbf{A},\mathbf{B})$ can be reduced to a pair $(\mathbf{H},\mathbf{B})$, where $\mathbf{H}$ has tridiagonal plus rank-one structure. In this paper we propose two fast algorithms for reducing the pair $(\mathbf{A},\mathbf{B})$ to Hessenberg-triangular form. The matrix pair $(\mathbf{A},\mathbf{B})$ has two spurious infinite eigenvalues, and if the leading coefficients of the interpolant are zero, there will also be other infinite eigenvalues. We propose tools for detecting when the leading coefficients of $p(z)$ are zero, and describe a procedure to deflate all of the infinite eigenvalues from the reduced matrix pair $(\mathbf{H},\mathbf{B})$, while still maintaining the tridiagonal plus rank-one structure of the resulting standard eigenvalue problem. Since fast $QR$ algorithms exist for such struc...

The thermofluoric behavior of poly(fluorenetolyldiphenylamine)–oxadiazole pair in a polymer matrix

Under UV irradiation, 1,4-bis(5-(4-octan-2-ylphenyl)-1,3,4-oxadiazol-2-yl)naphthalene (NOXD) and poly(fluorenetolyldiphenylamine) (PFT) are blue light emissive in solid film. When NOXD and PFT were blended to form a neat thin-film, yellowish exciplex emission was observed. The emissive properties vary when the materials were blended into different kinds of polymeric hosts; while polystyrene–NOXD–PFT (5 : 1 : 1) and PMMA–NOXD–PFT (5 : 1 : 1) showed blue light emission, poly(acrylonitrile-co-methyl acrylate) (P(AN-co-MA))–NOXD–PFT (5 : 1 : 1) showed blue and yellow dual emissions. When the film was heated at 140 °C for 2 min, the yellow light exciplex emission was enhanced. However, upon cooling to ambient temperature, the exciplex emission intensity gradually dropped back. Similar thermofluoric behaviors were observed when the thermally crosslinked polyepoxy-polymercaptan–NOXD–PFT film (20 : 1 : 1) was heated at 160–170 °C; the blue light emissive film showed yellow light emission, and turned white when cooling down to ambient temperature. This thermofluoric phenomenon is recyclable. We attributed the observation of the yellow emission to the segregation of NOXD from the polymeric host at high temperature that allows NOXD to aggregate with PFT, leading to exciplex emission.

A note on Frobenius norm preservers of Jordan product

[1] A. FOSNER, B. KUZMA, T. KUZMA, N.-S. SZE, Maps preserving matrix pairs with zero Jordan product, Linear Multilinear Algebra 59, 5 (2011), 507–529. [2] R. HORN, C. JOHNSON, Topics in Matrix Analysis. Cambridge UP, 1991. [3] B. KUZMA, G. LESNJAK, C.-K. LI, T. PETEK, L. RODMAN, Conditions for linear independence of two operators. Operator Theory: Advances and Applications 202 (2010), 411–434. [4] B. KUZMA, G. LESNJAK, C.-K. LI, T. PETEK, L. RODMAN, Norm preservers of Jordan products, Electron. J. Linear Algebra 22 (2011), 959–978.

The product field of values

For two n-by-n matrices, A,B, the product field of values is the set P(A,B)={〈Ax,x〉〈Bx,x〉:x∈Cn,||x||=1}. In this paper, we establish basic properties of the product field of values. The main results are a proof that the product field is a simply connected subset of the complex plane and a characterization of matrix pairs for which the product field has nonempty interior.

Two-sided generalized hyperbolic QR factorization and its perturbation analysis

We first propose the two-sided generalized hyperbolic QR factorization of a matrix pair, which is a generalization of the generalized QR factorization. More explicitly, two orthogonal matrices in the generalized QR factorization are replaced with two corresponding J-orthogonal matrices, where J is a signature matrix. Then, we consider the perturbation analysis of the new matrix factorization. Some first order normwise perturbation bounds are derived using the refined matrix equation approach and the updated matrix–vector equation approach. The corresponding perturbation bounds for the generalized QR factorization are also obtained as the special case. In comparison, these bounds can be much tighter than the previous ones derived from Barrlund (1994) [A. Barrlund, Perturbation bounds for the generalized QR factorization, Linear Algebra Appl. 207 (1994) 251–271].

Optimal Approximate Solution of a Class of Matrix Equation

The optimal approximate solution of a class of matrix equation is considered. Based on the generalized singular value decomposition, the expression of the optimal approximate solution to a given matrix pair is derived in the symmetric solution set.

On the generalized reflexive and anti-reflexive solutions to a system of matrix equations

Let P and Q be two generalized reflection matrices, i.e, P=PH, P2=I and Q=QH, Q2=I. An n×n matrix A is said to be generalized reflexive (generalized anti-reflexive) with respect to the matrix pair (P;Q) if A=PAQ(A=-PAQ). It is obvious that any n×m matrix is also a generalized reflexive with respect to the matrix pair (In;Im). By extending the conjugate gradient least square (CGLS) approach, the present paper treats two iterative algorithms to solve the system of matrix equationsF1(X)=A1,F2(X)=A2,⋮⋮⋮Fm(X)=Am,(including the Sylvester and Lyapunov matrix equations as special cases) over the generalized reflexive and anti-reflexive matrices, where F1,F2,…,Fm are the linear operators from Cn×n onto Cri×si and Ai∈Cri×si for i=1,2,…,m. When this system is consistent over the generalized reflexive (generalized anti-reflexive) matrix, it is proved that the first (second) iterative algorithm converges to a generalized reflexive (generalized anti-reflexive) solution for any initial generalized reflexive (generalized anti-reflexive) matrix. Also the first (second) iterative algorithm can obtain the the least Frobenius norm generalized reflexive (generalized anti-reflexive) solution for special initial generalized reflexive (generalized anti-reflexive) matrix. Furthermore, the optimal approximation generalized reflexive (generalized anti-reflexive) solution to a given generalized reflexive (generalized anti-reflexive) matrix can be derived by finding the least Frobenius norm generalized reflexive (generalized anti-reflexive) solution of a new system of matrix equations. Finally, we test the proposed iterative algorithms and show their effectiveness using numerical examples.

A Hoffman–Wielandt-type residual bound for generalized eigenvalues of a definite pair

Let {A,B} be a definite matrix pair of order n, and let the columns of X˜1∈Cn×l span an l-dimensional approximate invariant eigenspace of {A,B}. In this paper, we present a Hoffman–Wielandt-type residual bound for eigenvalues between Rayleigh quotient matrix pair {X˜1HAX˜1,X˜1HBX˜1} and {A,B}, which extends a result of Kahan [W. Kahan, Numerical Linear Algebra, Canadian Mathematical Bulletin, 9 (1966) 757–801].

Low-temperature photoluminescence of ion-implanted SiO2:Sn+ films and glasses

Low-temperature photoluminescence spectroscopy with pulsed synchrotron excitation is applied to study the regularities of excitation and relaxation of both point defects and nanoparticles formed by tin implantation into SiO2 films and glasses. It has been found that tin implantation followed by air and nitrogen annealing yields the formation of α-Sn nanoclusters and nonstoichiometric SnO x nanoparticles, while a stable phase of SnO2 does not appear. Alternative channels of luminescence excitation are revealed for nanoclusters, including energy transfer from excitons and electron-hole pairs of the host SiO2 matrix.

Fluorescence Characteristics of Isolated Dye Molecules within Silicalite-1 Channels

Fluorescence characteristics of hemicyanine dye molecules isolated from neighboring molecules and strongly restricted inside nanosized pores of zeolite (silicalite-1) crystal were investigated. For samples in which the molecules were sufficiently far away from the others, the fluorescence decay lifetime of the molecules was about 2.2 ns. As the intermolecular distance was reduced, the steady-state fluorescence peak shifted toward the longer wavelength and the fluorescence efficiency decreased markedly. The fluorescence decay lifetime also decreased to 0.8 ns for a sample with the smallest intermolecular distance of 2.1 nm. These results were explained in terms of a dipole-dipole interaction between pairs of dye molecules. From the relation between the intermolecular distances and the fluorescence decay lifetimes of the molecules, the radius of energy transfer of hemicyanine donor-acceptor pair in zeolite matrix was determined to be 2.2 nm, in fair agreement with the calculated Förster radius between dye molecules of the same species.

Routing and wavelength assignment in optical networks using Artificial Bee Colony algorithm

This paper presents an intelligent approach for modelling Routing and wavelength assignment (RWA) problem in wavelength-routed Dense-Wavelength-Division-Multiplexing (DWDM) optical networks. A new idea based on Artificial Bee Colony (ABC) algorithm is introduced for solving RWA problem which is known to be an NP-hard problem. In the proposed ABC-RWA approach every food source represents a possible and feasible candidate lightpath between each original and destination node pair in demand matrix. The positions of food sources are modified by some artificial bees in the population where the aim is to discover the places of food sources. The food source with the highest nectar value seems to be a solution which is evaluated by the fitness function. The proposed approach is evaluated for both path length (propagation delay) and hops count optimization schemes for PAN EUROPEAN and NSFNET test bench optical networks. The performance of ABC based approach is compared with the Genetic Algorithm (GA) model for solving RWA problem under random and heavy load traffic models. Simulation results demonstrate the ability and efficiency of proposed ABC model for solving RWA in real-world optical networks. Furthermore a comparison study approves that ABC is faster than GA to hit RWA global optimization solutions due to less complexity and computational processing.

Sensitivity analysis for the generalized singular value decomposition

SUMMARYIn this paper, we discuss the sensitivity of multiple nonzero finite generalized singular values and the corresponding generalized singular matrix set of a real matrix pair analytically dependent on several parameters. From our results, the partial derivatives of multiple nonzero singular values and their left and right singular vector matrices are obtained.Copyright © 2012 John Wiley & Sons, Ltd.

The rotation of eigenspaces of perturbed matrix pairs

We present new sinΘ theorems for perturbations of positive definite matrix pairs. The rotation of eigenspaces is measured in the matrix dependent scalar product. We assess the sharpness of the new estimates in terms of effectivity quotients (the quotient of the measure of the perturbation and the estimator). Known relative sinΘ theorems for eigenspaces of positive definite Hermitian matrices are included as special cases in our approach. Our experiments indicate that relative sinΘ theorems are asymptotically sharp when the rotation is measured in the appropriate matrix dependent scalar product but not always in the ordinary Euclidean scalar product.

Computing the radius of controllability for state space systems

In this paper, we discuss the problem of computing the nearest uncontrollable system to a given control system represented by a matrix pair ( A , B ) . In order to do so, we construct a sequence of structured matrices from given system matrices A and B . Controllability of the pair ( A , B ) is equivalent to a condition on the null-space dimension of an appropriate matrix in this sequence. We show that the dimension of the reachability space is also related to the above condition. Further, it is shown that the nearest Structured Low Rank Approximation (SLRA) of this structured matrix corresponds to a nearest uncontrollable system to the pair ( A , B ) .

On “P” property and the column-W property

P-matrices play an important role in the well-posedness of a linear complementarity problem (LCP). Similarly, the well-posedness of a horizontal linear complementarity problem (HLCP) is closely related to the column-W property of a matrix k-tuple.In this paper we first consider the problem of generating P-matrices from a given pair of matrices. Given a matrix pair (D,F) where D is a square matrix of order m and matrix F has m rows, “what are the conditions under which there exists a matrix G such that (D+FG) is a P-matrix?”. We obtain necessary and sufficient conditions for the special case when the column rank of F is m−1. A decision algorithm of complexity O(m2) to check whether the given pair of matrices (D,F) is P-matrisable is obtained. We also obtain a necessary and an independent sufficient condition for the general case when rank(F) is less than m−1.We then generalise the P-matrix generating problem to the generation of matrix k-tuples satisfying the column-W property from a given matrix (k+1)-tuple. That is, given a matrix (k+1)-tuple (D1,…,Dk,F), where Djs are square matrices of order m and F is a matrix having m rows, we determine the conditions under which the matrix k-tuple (D1+FG1,…,Dk+FGk) satisfies the column-W property. As in the case of P-matrices we obtain necessary and sufficient conditions for the case when rank(F)=m−1. Using these conditions a decision algorithm of complexity O(km2) to check whether the given matrix (k+1)-tuple is column-W matrisable is obtained. Then for the case when rank(F) is less than m−1, we obtain a necessary and an independent sufficient condition.For a special sub-class of P-matrices we give a polynomial time decision algorithm for P-matrisability. Finally, we obtain a geometric characterisation of column-W property by generalising the well known separation theorem for P-matrices.

Classification of the entangled statesL×N×N

We presented a general classification scheme for the tripartite $L\ifmmode\times\else\texttimes\fi{}N\ifmmode\times\else\texttimes\fi{}N$ entangled system under stochastic local operation and classical communication. The whole classification procedure consists of two correlated parts: the simultaneous similarity transformation of a commuting matrix pair into a canonical form and the study of internal symmetry of parameters in the canonical form. Based on this scheme, a concrete example of entanglement classification for a $3\ifmmode\times\else\texttimes\fi{}N\ifmmode\times\else\texttimes\fi{}N$ system is given.

An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations (AXB − CYD, EXF − GYH) = (M, N), which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matrices X and Y. When the matrix equations are consistent, for any initial generalized reflexive matrix pair [X1, Y1], the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round‐off errors, and the least Frobenius norm generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair to a given matrix pair [X0, Y0] in Frobenius norm can be derived by finding the least‐norm generalized reflexive solution pair of a new corresponding generalized coupled Sylvester matrix equation pair , where . Several numerical examples are given to show the effectiveness of the presented iterative algorithm.