# Linear Algebra and its Applications

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Balancing control and simplicity: A variable aggregation method in intensity modulated radiation therapy planning
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April 2008

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It is commonly believed that not all degrees of freedom are needed to produce good solutions for the treatment planning problem in intensity modulated radiation therapy (IMRT). However, typical methods to exploit this fact either increase the complexity of the optimization problem or are heuristic in nature. In this work we introduce a technique based on adaptively refining variable clusters to successively attain better treatment plans. The approach creates approximate solutions based on smaller models that may come arbitrarily close to the optimal solution. Although the method is illustrated using a specific treatment planning model, the components constituting the variable clustering and the adaptive refinement are independent of the particular optimization problem.

On Linear Infeasibility Arising in Intensity-Modulated Radiation Therapy Inverse Planning
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March 2008

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Intensity-modulated radiation therapy (IMRT) gives rise to systems of linear inequalities, representing the effects of radiation on the irradiated body. These systems are often infeasible, in which case one settles for an approximate solution, such as an {α, β}-relaxation, meaning that no more than α percent of the inequalities are violated by no more than β percent. For real-world IMRT problems, there is a feasible {α, β}-relaxation for sufficiently large α, β > 0, however large values of these parameters may be unacceptable medically.The {α, β}-relaxation problem is combinatorial, and for given values of the parameters can be solved exactly by Mixed Integer Programming (MIP), but this may be impractical because of problem size, and the need for repeated solutions as the treatment progresses.As a practical alternative to the MIP approach we present a heuristic non-combinatorial method for finding an approximate relaxation. The method solves a Linear Program (LP) for each pair of values of the parameters {α, β} and progresses through successively increasing values until an acceptable solution is found, or is determined non-existent. The method is fast and reliable, since it consists of solving a sequence of LP's.

A lower bound on the error in dimensionality reduction resulting from projection onto a restricted subspace

December 2010

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We obtain the lower bound on a variant of the common problem of dimensionality reduction. In this version, the dataset is projected on to a k dimensional subspace with the property that the first k-1 basis vectors are fixed, leaving a single degree of freedom in terms of basis vectors.

Design, parametrization, and pole placement of stabilizing output feedback compensators via injective cogenerator quotient signal modules

March 2012

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Control design belongs to the most important and difficult tasks of control engineering and has therefore been treated by many prominent researchers and in many textbooks, the systems being generally described by their transfer matrices or by Rosenbrock equations and more recently also as behaviors. Our approach to controller design uses, in addition to the ideas of our predecessors on coprime factorizations of transfer matrices and on the parametrization of stabilizing compensators, a new mathematical technique which enables simpler design and also new theorems in spite of the many outstanding results of the literature: (1) We use an injective cogenerator signal module ℱ over the polynomial algebra [Formula: see text] (F an infinite field), a saturated multiplicatively closed set T of stable polynomials and its quotient ring [Formula: see text] of stable rational functions. This enables the simultaneous treatment of continuous and discrete systems and of all notions of stability, called T-stability. We investigate stabilizing control design by output feedback of input/output (IO) behaviors and study the full feedback IO behavior, especially its autonomous part and not only its transfer matrix. (2) The new technique is characterized by the permanent application of the injective cogenerator quotient signal module [Formula: see text] and of quotient behaviors [Formula: see text] of [Formula: see text]-behaviors B. (3) For the control tasks of tracking, disturbance rejection, model matching, and decoupling and not necessarily proper plants we derive necessary and sufficient conditions for the existence of proper stabilizing compensators with proper and stable closed loop behaviors, parametrize all such compensators as IO behaviors and not only their transfer matrices and give new algorithms for their construction. Moreover we solve the problem of pole placement or spectral assignability for the complete feedback behavior. The properness of the full feedback behavior ensures the absence of impulsive solutions in the continuous case, and that of the compensator enables its realization by Kalman state space equations or elementary building blocks. We note that every behavior admits an IO decomposition with proper transfer matrix, but that most of these decompositions do not have this property, and therefore we do not assume the properness of the plant. (4) The new technique can also be applied to more general control interconnections according to Willems, in particular to two-parameter feedback compensators and to the recent tracking framework of Fiaz/Takaba/Trentelman. In contrast to these authors, however, we pay special attention to the properness of all constructed transfer matrices which requires more subtle algorithms.

Solution of a Complex Least Squares Problem with Constrained Phase

December 2010

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The least squares solution of a complex linear equation is in general a complex vector with independent real and imaginary parts. In certain applications in magnetic resonance imaging, a solution is desired such that each element has the same phase. A direct method for obtaining the least squares solution to the phase constrained problem is described.

On the nullity of a graph with cut-points

January 2012

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Let G be a simple graph of order n and A(G) be its adjacency matrix. The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in the spectrum of A(G). Denote by Ck and Lk the set of all connected graphs with k induced cycles and the set of line graphs of all graphs in Ck, respectively. In 1998, Sciriha [I. Sciriha, On singular line graphs of trees, Congr. Numer. 135 (1998) 73-91] show that the order of every tree whose line graph is singular is even. Then Gutman and Sciriha [I. Gutman, I. Sciriha, On the nullity of line graphs of trees, Discrete Math. 232 (2001) 35-45] show that the nullity set of L0 is {0,1}. In this paper, we investigate the nullity of graphs with cut-points and deduce some concise formulas. Then we generalize Scirihas' result, showing that the order of every graph G is even if such a graph G satisfies that G∈Ck and η(L(G))=k+1, and the nullity set of Lk is {0,1,…,k,k+1} for any given k, where L(G) denotes the line graph of the graph G.

Discrete Fourier transform of prime order: Eigenvectors with small support

March 2013

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We show how to construct an eigenvector basis of the discrete Fourier transform of odd prime order. The special feature of the new basis is that the basis vectors have small support.

On the Nonnegative Rank of Euclidean Distance Matrices

September 2010

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The Euclidean distance matrix for n distinct points in ℝ r is generically of rank r + 2. It is shown in this paper via a geometric argument that its nonnegative rank for the case r = 1 is generically n.

A fast algorithm for solving a linear feasibility problem with application to Intensity-Modulated Radiation Therapy

April 2008

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The goal of Intensity-Modulated Radiation Therapy (IMRT) is to deliver sufficient doses to tumors to kill them, but without causing irreparable damage to critical organs. This requirement can be formulated as a linear feasibility problem. The sequential (i.e., iteratively treating the constraints one after another in a cyclic fashion) algorithm ART3 is known to find a solution to such problems in a finite number of steps, provided that the feasible region is full dimensional. We present a faster algorithm called ART3+. The idea of ART3+ is to avoid unnecessary checks on constraints that are likely to be satisfied. The superior performance of the new algorithm is demonstrated by mathematical experiments inspired by the IMRT application.

Distribution Metrics and Image Segmentation

October 2007

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The purpose of this paper is to describe certain alternative metrics for quantifying distances between distributions, and to explain their use and relevance in visual tracking. Besides the theoretical interest, such metrics may be used to design filters for image segmentation, that is for solving the key visual task of separating an object from the background in an image. The segmenting curve is represented as the zero level set of a signed distance function. Most existing methods in the geometric active contour framework perform segmentation by maximizing the separation of intensity moments between the interior and the exterior of an evolving contour. Here one can use the given distributional metric to determine a flow which minimizes changes in the distribution inside and outside the curve.

The discrete algebraic Riccati equation and linear matrix inequality

January 1995

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We study the discrete time algebraic Riccati equation. In particular we show that even in the most general cases there exists a one-one correspondence between solutions of the algebraic Riccati equation and deflating subspaces of a matrix pencil. We also study the relationship between algebraic Riccati equation and the discrete time linear matrix inequality. We show that in general only a subset of the set of rank-minimizing solutions of the linear matrix inequality correspond to the solutions of the associated algebraic Riccati equation, and study under what conditions these sets are equal. In this process we also derive very weak assumptions under which a Riccati equation has a solution.

LMI characterization of structural and robust stability

February 1999

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Introduces several stability conditions for a given class of matrices expressed in terms of linear matrix inequalities, being thus simply and efficiently computable. Diagonal and simultaneous stability, both characterized by polytopes of matrices, are addressed. Using this approach a method particularly attractive to test a given matrix for D-stability is proposed. Lyapunov parameter dependent functions are built in order to reduce conservativeness of the stability conditions. The key idea is to relate Hurwitz stability with a positive realness condition

On minimizing the largest eigenvalue of a symmetric matrix

February 1992

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The problem of minimizing the largest eigenvalue over an affine family of symmetric matrices is considered. This problem has a variety of applications, such as the stability analysis of dynamic systems or the computation of structured singular values. Given ∈&ges;0, an optimality condition is given which ensures that the largest eigenvalue is within ∈ error bound of the solution. A novel line search rule is proposed and shown to have good descent property. When the multiplicity of the largest eigenvalue at solution is known, a novel algorithm for the optimization problem under consideration is derived. Numerical experiments show that the algorithm has good convergence behavior

State for 2-D systems

September 1989

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A new definition of state for N-D systems is given in a noncausal context. This definition is based on a deterministic Markovian-like property. It is shown that, for the particular case of (AR) 2-D systems, it yields systems that can be described by a special kind of first-order equations. The solutions of these equations can be simulated by means of a local line-by-line computational scheme.

Matrix measures, moment spaces and Favard's theorem for the interval [0,1] and [0,∞)

April 2002

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In this paper we study the moment spaces corresponding to matrix measures on compact intervals and on the nonnegative line [0,∞). A representation for nonnegative definite matrix polynomials is obtained, which is used to characterize moment points by properties of generalized Hankel matrices. We also derive an explicit representation of the orthogonal polynomials with respect to a given matrix measure, which generalizes the classical determinant representation of the one-dimensional case. Moreover, the coefficients in the recurrence relations can be expressed explicitly in terms of the moments of the matrix measure. These results are finally used to prove a refinement of the well-known Favard theorem for matrix measures, which characterizes the domain of the underlying measure of orthogonality by properties of the coefficients in the recurrence relationships.

Minimal nonnegative integral matrices and uniquely determined (0,1)-matrices

January 2002

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In this note we give an elementary proof of a theorem that characterizes those three-dimensional (0,1)-matrices that are determined by their plane sum vectors.

Tridiagonal pairs of shape (1,2,1)

July 2008

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Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A:V→V and A∗:V→V that satisfies the following conditions: (i) each of A,A∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A∗Vi⊆Vi-1+Vi+Vi+1 for 0⩽i⩽d, where V-1=0 and Vd+1=0; (iii) there exists an ordering of the eigenspaces of A∗ such that for 0⩽i⩽δ, where and ; (iv) there is no subspace W of V such that AW⊆W, A∗W⊆W, W≠0,W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and that for 0⩽i⩽d the dimensions of coincide; we denote this common value by ρi. The sequence is called the shape of the pair. In this paper we assume the shape is (1,2,1) and obtain the following results. We describe six bases for V; one diagonalizes A, another diagonalizes A∗, and the other four underlie the split decompositions for A,A∗. We give the action of A and A∗ on each basis. For each ordered pair of bases among the six, we give the transition matrix. At the end we classify the tridiagonal pairs of shape (1,2,1) in terms of a sequence of scalars called the parameter array.

Ilić, A.: Distance spectra and distance energy of integral circulant graphs. Linear Algebra Appl. 433, 1005-1014

October 2010

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The distance energy of a graph G is a recently developed energy-type invariant, defined as the sum of absolute values of the eigenvalues of the distance matrix of G. There was a vast research for the pairs and families of non-cospectral graphs having equal distance energy, and most of these constructions were based on the join of graphs. A graph is called circulant if it is Cayley graph on the circulant group, i.e. its adjacency matrix is circulant. A graph is called integral if all eigenvalues of its adjacency matrix are integers. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer. In this paper, we characterize the distance spectra of integral circulant graphs and prove that these graphs have integral eigenvalues of distance matrix D. Furthermore, we calculate the distance spectra and distance energy of unitary Cayley graphs. In conclusion, we present two families of pairs (G1,G2) of integral circulant graphs with equal distance energy – in the first family G1 is subgraph of G2, while in the second family the diameter of both graphs is three.

On the polynomial identities of the algebra $M_{11}(E)$

June 2013

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Verbally prime algebras are important in PI theory. They were described by Kemer over a field $K$ of characteristic zero: 0 and $K$ (the trivial ones), $M_n(K)$, $M_n(E)$, $M_{ab}(E)$. Here $K$ is the free associative algebra of infinite rank, with free generators $T$, $E$ denotes the infinite dimensional Grassmann algebra over $K$, $M_n(K)$ and $M_n(E)$ are the $n\times n$ matrices over $K$ and over $E$, respectively. The algebras $M_{ab}(E)$ are subalgebras of $M_{a+b}(E)$, see their definition below. The generic (also called relatively free) algebras of these algebras have been studied extensively. Procesi described the generic algebra of $M_n(K)$ and lots of its properties. Models for the generic algebras of $M_n(E)$ and $M_{ab}(E)$ are also known but their structure remains quite unclear. In this paper we study the generic algebra of $M_{11}(E)$ in two generators, over a field of characteristic 0. In an earlier paper we proved that its centre is a direct sum of the field and a nilpotent ideal (of the generic algebra), and we gave a detailed description of this centre. Those results were obtained assuming the base field infinite and of characteristic different from 2. In this paper we study the polynomial identities satisfied by this generic algebra. We exhibit a basis of its polynomial identities. It turns out that this algebra is PI equivalent to a 5-dimensional algebra of certain upper triangular matrices. The identities of the latter algebra have been studied; these were described by Gordienko. As an application of our results we describe the subvarieties of the variety of unitary algebras generated by the generic algebra in two generators of $M_{11}(E)$. Also we describe the polynomial identities in two variables of the algebra $M_{11}(E)$.

Pang, J.-S.: Error bounds for the linear complementarity problem with a P-matrix. Linear Algebra Appl. 132, 123-136

April 1990

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We introduce a fundamental quantity associated with a P-matrix and show how this quantity is useful in deriving error bounds for the linear complementarity problem of the P-type. We also obtain (upper and lower) bounds for the quantity introduced.

Some properties of [tr(Q2p)]12p with application to linear minimax estimation

December 1990

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A nondifferentiable minimization problem is considered which occurs in linear minimax estimation. This problem is solved by replacing the nondifferentiable maximal eigenvalue of a real nonnegative definite matrix Q with [tr(Q2p)]1/2p. It is shown that any descent algorithm with inexact step-length rule can be used to obtain linear minimax estimators for the parameter vector of a parameter-restricted linear model.

Dür M.: Algorithmic copositivity detection by simplicial partition. Linear Algebra Appl. 428, 1511-1523

April 2008

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We present new criteria for copositivity of a matrix, i.e., conditions which ensure that the quadratic form induced by the matrix is nonnegative over the nonnegative orthant. These criteria arise from the representation of the quadratic form in barycentric coordinates with respect to the standard simplex and simplicial partitions thereof. We show that, as the partition gets finer and finer, the conditions eventually capture all strictly copositive matrices. We propose an algorithmic implementation which considers several numerical aspects. As an application, we present results on the maximum clique problem. We also briefly discuss extensions of our approach to copositivity with respect to arbitrary polyhedral cones.

Sir Thomas Muir, 1844–1934

December 2005

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Part I of this paper on Sir Thomas Muir, deals with his life in Scotland and at the Cape of Good Hope. In 1892, Thomas Muir, mathematician and educator, became the third Superintendent General of Education in the Cape Colony under British rule. He will be remembered as one of the greatest organisers and reformers in the history of Cape education. Muir found relief from his arduous administrative duties by his investigations in the field of mathematics, and, in particular, of algebra. Most of his more than 320 papers were on determinants and allied subjects. His magnum opus was a five-volume work: The Theory of Determinants in the Historical Order of Development (London, 1890–1930). Muir’s publications will be covered in Part II of this paper. However, a treatment of the contents of Muir’s papers and his vast contribution to the theory of determinants, fall beyond the scope of this paper.

On two perturbation estimates of the extreme solutions to the equations X±A*X−1A=Q

February 2006

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Two perturbation estimates for maximal positive definite solutions of equations X+A*X-1 A = Q and X - A*X(-1)A = Q are considered. These estimates are proved in [Hasanov et al., Improved perturbation Estimates for the Matrix Equations X A*X(-1)A = Q,LinearAlgebra Appl. 379 (2004) 113-135]. We derive new perturbation estimates under weaker restrictions on coefficient matrices of the equations. The theoretical results are illustrated by numerical examples. (c) 2005 Elsevier Inc. All rights reserved.

Perturbation analysis of the maximal solution of the matrix equation X+A∗X−1A=P. II

March 2003

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Consider the nonlinear matrix equationX+A∗X−1A=P,where A,P are n×n complex matrices with P Hermitian positive definite, and A* denotes the conjugate transpose of a matrix A. In this paper a sharper perturbation bound for the maximal solution to the matrix equation is derived, explicit expressions of the condition number for the maximal solution are obtained, and the backward error of an approximate solution to the maximal solution is evaluated by using the techniques developed in [Linear Algebra Appl. 259 (1997) 183; Linear Algebra Appl. 350 (2002) 237]. The results are illustrated by using numerical examples.

Perturbation analysis of the maximal solution of the matrix equation X+A∗X−1A=P

October 2001

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The Silences of the Archives, the Reknown of the Story. The Martin Guerre affair has been told many times since Jean de Coras and Guillaume Lesueur published their stories in 1561. It is in many ways a perfect intrigue with uncanny resemblance, persuasive deception and a surprizing end when the two Martin stood face to face, memory to memory, before captivated judges and a guilty feeling Bertrande de Rols. The historian wanted to go beyond the known story in order to discover the world of the heroes. This research led to disappointments and surprizes as documents were discovered concerning the environment of Artigat’s inhabitants and bearing directly on the main characters thanks to notarial contracts. Along the way, study of the works of Coras and Lesueur took a new direction. Coming back to the affair a quarter century later did not result in finding new documents (some are perhaps still buried in Spanish archives), but by going back over her tracks, the historian could only be struck by the silences of the archives that refuse to reveal their secrets and, at the same time, by the possible openings they suggest, by the intuition that almost invisible threads link here and there characters and events.

Groups generated by two elliptic elements in PU(2,1)

December 2010

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et f and g be two elliptic elements in PU(2,1). We prove that if the distance δ(f,g) between the complex lines or points fixed by f and g is large than a certain number, then the group 〈f,g〉 is discrete non-elementary and isomorphic to the free product 〈f〉∗〈g〉.

Freund, R.W.: On Padé-type model order reduction of J-Hermitian linear dynamical systems. Linear Algebra Appl. 429, 2451-2464

November 2008

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A simple, yet powerful approach to model order reduction of large-scale linear dynamical systems is to employ projection onto block Krylov subspaces. The transfer functions of the resulting reduced-order models of such projection methods can be characterized as Padé-type approximants of the transfer function of the original large-scale system. If the original system exhibits certain symmetries, then the reduced-order models are considerably more accurate than the theory for general systems predicts. In this paper, the framework of J-Hermitian linear dynamical systems is used to establish a general result about this higher accuracy. In particular, it is shown that in the case of J-Hermitian linear dynamical systems, the reduced-order transfer functions match twice as many Taylor coefficients of the original transfer function as in the general case. An application to the SPRIM algorithm for order reduction of general RCL electrical networks is discussed.

The Derivative of an Orthogonal Matrix of Eigenvectors of a Symmetric Matrix, Linear Algebra and its Applications, 264, 489-493

October 1997

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The authors supply the derivative of an orthogonal matrix of eigenvectors of a real symmetric matrix. To illustrate the applicability of their result they consider a real symmetric random matrix for which a more or less standard convergence in distribution is assumed to hold. The well-known delta method is then used to get the asymptotic distribution of the orthogonal eigenmatrix of the random matrix.

Properties of positive definite solutions of the equation X + A∗X−2A = I

August 1998

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In this paper we discuss some properties of a positive definite solution of the matrix equation X + A∗X−2 A = I. Two effective iterative methods for computing a positive definite solution of this equation are proposed. Necessary and sufficient conditions for existence of a positive definite solution are derived. Numerical experiments are executed with these methods.

The association schemes of dual polar spaces of type 2A2d-1(pf) are characterized by their parameters if d⩾3

March 1989

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It is shown that the cliques in a distance-regular graph Γ whose parameters are that of the dual polar space graph of type have size pf + 1. This means that Γ is the point graph of a regular near 2d-gon. If d = 2, we obtain the well-known result due to P.J. Cameron, J.M. Goethals, and J.J. Seidel that a pseudogeometric graph with parameters of the point graph of a generalized quadrangle of type (q,q2) is geometric. If d⩾3, then some results due to P.J. Cameron, E.E. Shult, A. Yanushka, and J. Tits on near 2d-gons imply that Γ coincides with the dual polar space graph of type .

Kronecker Product based Stability Tests and Performance Bounds for a Class of 2D Continuous-Discrete Linear Systems

September 2002

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This paper reports further development of the so-called 1D Lyapunov equation based approach to the stability analysis of differential linear repetitive processes which are a distinct class of 2D continuous–discrete linear systems of both practical and theoretical interest. In particular, it is shown that this approach leads to stability tests which can be implemented by computations with matrices which have constant entries. Also if the example under consideration is stable then physically meaningful information concerning one key aspect of transient performance is available for no extra cost.

On the spectral and combinatorial structure of 2D positive systems

September 1996

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The dynamics of a 2D positive system depends on the pair of nonnegative square matrices that provide the updating of its local states. In this paper, several spectral properties, such as finite memory, separability, and property L, which depend on the characteristic polynomial of the pair, are investigated under the nonnegativity constraint and in connection with the combinatorial structure of the matrices. Some aspects of the Perron-Frobenius theory are extended to the 2D case; in particular, conditions are provided guaranteeing the existence of a common maximal eigenvector for two nonnegative matrices with irreducible sum. Finally, some results on 2D positive realizations are presented.

Homogeneous geodesics in the flag manifold SO(2l+1)/U(l-m)×SO(2m+1)

February 2008

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A geodesic curve in a Riemannian homogeneous manifold (M=G/K,g) is called a homogeneous geodesic if it is an orbit of an one-parameter subgroup of the Lie group G. We investigate G-invariant metrics such that all geodesics are homogeneous for the flag manifold M=SO(2l+1)/U(l-m)×SO(2m+1). By reformulating the problem into a matrix form we show that SO(2ℓ+1)/U(ℓ-m)×SO(2m+1) has homogeneous geodesics with respect to any SO(2ℓ+1)-invariant metric if and only if m=0. In all other cases this space admits at least one non-homogeneous geodesic. We also give examples of finding homogeneous geodesics in the above flag manifold for special values of l and m.

A note on the eigenvectors of Hadamard matrices of order 2n

June 1982

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A set of 2n mutually orthogonal eigenvectors of Hadamard matrices of order 2n are explicitly derived, and their properties are developed. An instance of Pell's equation driven by the Thue-Morse sequence is noted.

On a Norm Compression Inequality for 2XN Partitioned Block Matrices

February 2007

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We conjecture the following so-called norm compression inequality for $2\times N$ partitioned block matrices and the Schatten $p$-norms: for $p\ge 2$, $$||({array}{cccc} A_1 & A_2 & ... & A_N B_1 & B_2 & >... & B_N {array})||_p \le ||({array}{cccc} ||A_1||_p & ||A_2||_p & ... & ||A_N||_p \ ||B_1||_p & ||B_2||_p & ... & ||B_N||_p {array})||_p$$ while for $1\le p\le 2$ the ordering of the inequality is reversed. This inequality includes Hanner's inequality for matrices as a special case. We prove several special cases of this inequality and give examples for $3\times 3$ and larger partitionings where it does not hold.

Slapničar, I.: Highly accurate symmetric eigenvalue decomposition and hyperbolic SVD. Linear Algebra Appl. 358, 387-424

January 2003

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Let G be a m×n real matrix with full column rank and let J be a n×n diagonal matrix of signs, . The hyperbolic singular value decomposition (HSVD) of the pair (G,J) is defined as G=UΣV−1, where U is orthogonal, Σ is positive definite diagonal, and V is J-orthogonal matrix, . We analyze when it is possible to compute the HSVD with high relative accuracy. This essentially means that each computed hyperbolic singular value is guaranteed to have some correct digits, even if they have widely varying magnitudes. We show that one-sided J-orthogonal Jacobi method method computes the HSVD with high relative accuracy. More precisely, let B=GD−1, where D is diagonal such that the columns of B have unit norms. Essentially, we show that the computed hyperbolic singular values of the pair (G,J) will have correct decimal digits, where ε is machine precision. We give the necessary relative perturbation bounds and error analysis of the algorithm. Our numerical tests confirmed all theoretical results.For the symmetric non-singular eigenvalue problem Hx=λx, we analyze the two-step algorithm which consists of factorization followed by the computation of the HSVD of the pair (G,J). Here G is square and non-singular. Let , where is diagonal such that the rows of have unit norms, and let B be defined as above. Essentially, we show that the computed eigenvalues of H will have correct decimal digits. This accuracy can be much higher then the one obtained by the classical QR and Jacobi methods applied to H, where the accuracy depends on the spectral condition number of H, particularly if the matrices B and are well conditioned, and we are interested in the accurate computation of tiny eigenvalues. Again, we give the perturbation and error bounds, and our theoretical predictions are confirmed by a series of numerical experiments.We also give the corresponding results for eigenvectors and hyperbolic singular vectors.

On graphs whose signless Laplacian index does not exceed 4.5

July 2009

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Let AG and DG be respectively the adjacency matrix and the degree matrix of a graph G. The signless Laplacian matrix of G is defined as QG=DG+AG. The Q-spectrum of G is the set of the eigenvalues together with their multiplicities of QG. The Q-index of G is the maximum eigenvalue of QG. The possibilities for developing a spectral theory of graphs based on the signless Laplacian matrices were discussed by Cvetković et al. [D. Cvetković, P. Rowlinson, S.K. Simić, Signless Laplacians of finite graphs, Linear Algebra Appl. 423 (2007) 155–171]. In the latter paper the authors determine the graphs whose Q-index is in the interval [0,4]. In this paper, we investigate some properties of Q-spectra of graphs, especially for the limit points of the Q-index. By using these results, we characterize respectively the structures of graphs whose the Q-index lies in the intervals , and (ϵ+2,4.5], where .

Brand, M.: Fast low-rank modifications of the thin singular value decomposition. Linear Algebra Appl. 415(1), 20-30

May 2006

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This paper develops an identity for additive modifications of a singular value decomposition (SVD) to reflect updates, downdates, shifts, and edits of the data matrix. This sets the stage for fast and memory-efficient sequential algorithms for tracking singular values and subspaces. In conjunction with a fast solution for the pseudo-inverse of a submatrix of an orthogonal matrix, we develop a scheme for computing a thin SVD of streaming data in a single pass with linear time complexity: A rank-r thin SVD of a p × q matrix can be computed in O(pqr) time for .

Sidiropoulos, N.D.: On Kruskal’s uniqueness condition for the Candecomp/Parafac decomposition. Lin. Alg. Appl. 420, 540-552

January 2007

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Let X be a real-valued three-way array. The Candecomp/Parafac (CP) decomposition is written as X = Y(1) + ⋯ + Y(R) + E, where Y(r) are rank-1 arrays and E is a rest term. Each rank-1 array is defined by the outer product of three vectors a(r), b(r) and c(r), i.e. . These vectors make up the R columns of the component matrices A, B and C. If 2R + 2 is less than or equal to the sum of the k-ranks of A, B and C, then the fitted part of the decomposition is unique up to a change in the order of the rank-1 arrays and rescaling/counterscaling of each triplet of vectors (a(r), b(r), c(r)) forming a rank-1 array. This classical result was shown by Kruskal. His proof is, however, rather inaccessible and does not seem intuitive. In order to contribute to a better understanding of CP uniqueness, this paper provides an accessible and intuitive proof of Kruskal’s condition. The proof is both self-contained and compact and can easily be adapted for the complex-valued CP decomposition.

Construction of a 45,694-dimensional simple module of Lyons' sporadic group over GF(2)

April 1997

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It is shown that the 2-modular reduction ϕ of the smallest nontrivial, rational, absolutely irreducible character X of Lyons' sporadic simple group Ly is irreducible. Furthermore, an explicit 45,694-dimensional irreducible matrix representation of Ly over the field GF(2) with two elements is given. The construction has been performed on a parallel computer of type IBM SP1 with 10 nodes. It uses the recent presentation of Ly as a permutation group on 9,606,125 points given by Cooperman, Finkelstein, Tselman, and York.

Equivalence after extension and matricial coupling coincide with Schur coupling, on separable Hilbert spaces Dedicated to Harm Bart on the occasion of his 70th birthday.

February 2013

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It is known that two Banach space operators that are Schur coupled are also equivalent after extension, or equivalently, matricially coupled. The converse implication, that operators which are equivalent after extension or matricially coupled are also Schur coupled, was only known for Fredholm Hilbert space operators and Fredholm Banach space operators with index 0. We prove that this implication also holds for Hilbert space operators with closed range, generalizing the result for Fredholm operators, and Banach space operators that can be approximated in operator norm by invertible operators. The combination of these two results enables us to prove that the implication holds for all operators on separable Hilbert spaces.

On the integer form of the additive Hilbert's Theorem 90

October 2004

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Let K be a number field whose ring of integers is a principal ideal domain and let E be a cyclic Galois extension of K. We prove that every integer of E with zero trace is a difference of two conjugates of an integer of E if and only if there is an integer of E with trace 1.

How many negative entries can A2 have?

March 1997

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A matrix whose entries are +, −, and 0 is called a sign pattern matrix. We first characterize sign patterns A such that A2 ⩽ 0. Further, we determine the maximum number of negative entries that can occur in A2 whenever A2 ⩽ 0, and then we characterize the sign patterns that achieve this maximum number. Next we find the maximum number of negative entries that can occur in the square of any sign pattern matrix, and provide a class of sign patterns that achieve this maximum. We also determine the maximum number of negative entries in the square of any real matrix. Finally, we discuss the spectral properties of the sign patterns whose squares contain the maximum number of negative entries in the special case when A2 ⩽ 0, and in the general case that includes any sign pattern.

A weak majorization involving the matrices A ∘ B and AB

July 1995

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Let A and B be n × n positive definite matrices, and let the eigenvalues of A ∘ B and AB be arranged in decreasing order. Then for all r > 0,Furthermore it is shown thatis a limiting case of the weak majorization above which settles a conjecture of Bapat and Johnson.

On almost nilpotent-by-abelian Lie algebras

November 1996

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We examine Lie algebras all of whose proper subalgebras are nilpotent-by-abelian but which themselves are not nilpotent-by-abelian. We study the existence and structure of these algebras.

Maximal Abelian subalgebras of pseudounitary lie algebras

July 1990

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The task of classifying and constructing all maximal Abelian subalgebras of su(p,q) (p⩾q⩾1) is reduced to that of classifying orthogonally indecomposable MASAs. These are either maximal Abelian nilpotent subalgebras (represented by nilpotent matrices in any finite-dimensional representation), or for p=q they can be (nonorthogonally) decomposable and their study can be reduced to a construction of MANSs of sl(p,C). Two types of MANSs of su(p,q) are shown to exist (“one-rowed” and “non-one-rowed”). Numerous classification theorems are proven and applied to obtain all MASAs of su(p,1), su(p,2), and su(p,q) with p+q⩽6. Physical applications are discussed.

Maximal Abelian Subalgebras of e(p,q) algebras

June 1998

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Maximal abclian subalgebras (MASAs) of one of the classical real inhomogencous Lie algebras are constructed, namely those of the pseudoeuclidean Lie algebra e(p, q). Use is made of the semidireet sum structure of e(p,q) with the translations T(p + q) as an abclian ideal. We first construct splitting MASAs that arc themselves direct sums of abelian subalgebras of o(p,q) and of subalgebras of T(p + q). The splitting subalgebras are used to construct the complementary nonsplitting ones. Here the results are less complete than in the splitting case. We present general decomposition theorems and construct indecomposable MASAs for all algebras e(p,q), p ≥ q ≥ 0. The case of q = 0 and 1 were treated earlier in a physical context. The case q = 2 is analyzed here in detail as an illustration of the general results.RésuméLes sous-algèbres maximales abéliennes (SAMAs) d’une algèbre réelle classique non-homogène sont construites, en particulier, celles d’algèbre de Lie pseudo-euclidienne e(p,q). On utilise la structure de la somme semi-directe de e(p,q) avec les translations T(p + q) qui représente un idéal abélien. Nous avons construit, en premier, les SAMAs “splitting”, qui sont des sommes directes des sous-algèbres abéliennes de o(p,q) et de sous-algèbres de T(p + q). Les sous-algèbres “splitting” sont utilisées pour construire les sous-algèbres complementaire “nonsplitting”. Les résultais ne sont pas explicites comme dans le cas des SAMAs “splitting”. Nous présentons les théorèmes généraux de décomposition et nous construisons les SAMAs indécomposables pour toutes les algèbres e(p. q), p ≥ q ≥ 0. Les cas de q = 0 et 1 sont déjà traités dans un context physique. Le cas q = 2 est analysé ici en détail comme une illustration des résultats généraux.

Integer invariants of abelian Cayley graphs

August 2013

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Let $G$ be a finite abelian group, let $E$ be a subset of $G$, and form the Cayley (directed) graph of $G$ with connecting set $E$. We explain how, for various matrices associated to this graph, the spectrum can be used to give information on the Smith normal form. This technique is applied to several interesting examples, including matrices in the Bose-Mesner algebra of the Hamming association scheme $H(n,q)$. We also recover results of Bai and Jacobson-Niedermaier-Reiner on the critical group of a Cartesian product of complete graphs.

Arc-transitive elementary abelian covers of the octahedron graph

October 2008

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