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Some preconditioners for accelerating the classical iterative methods are given in Zhang et al. [Y. Zhang and T.Z. Huang, A class of optimal preconditioners and their applications, Proceedings of the Seventh International Conference on Matrix Theory and Its Applications in China, 2006. Y. Zhang, T.Z. Huang, and X.P. Liu, Modified iterative methods for nonnegative matrices and M-matrices linear systems, Comput. Math. Appl. 50 (2005), pp. 1587-1602. Y. Zhang, T.Z. Huang, X.P. Liu, A class of preconditioners based on the (I+S(α))-type preconditioning matrices for solving linear systems, Appl. Math. Comp. 189 (2007), pp. 1737-1748]. Another kind of preconditioners approximating the inverse of a symmetric positive definite matrix was given in Simons and Yao [G. Simons, Y. Yao, Approximating the inverse of a symmetric positive definite matrix, Linear Algebra Appl. 281 (1998), pp. 97-103]. Zhang et al. 's preconditioners and Simons and Yao's are generalized in this paper. These preconditioners are all of low construction cost, which all could be taken as approximate inverse of M-matrices. Numerical experiments of these preconditioners applied with Krylov subspace methods show the effectiveness and performance, which also show that the preconditioners proposed in this paper are better approximate inverse for M-matrices than Simons'. [ABSTRACT FROM AUTHOR]

We study high order numerical quadratures to one dimensional delta function integrals in this paper. This is motivated by the fact that traditional numerical quadratures give only first order accuracy in general. We provide criteria on discrete delta functions and support size formulas which ensure any desired accuracy of the numerical quadratures. Discrete delta functions and support size formulas satisfying these criteria are designed. Numerical examples are presented to verify the performed analysis and the high order accuracy of the proposed numerical quadratures. [ABSTRACT FROM AUTHOR]

This paper discusses the eigenvalue problem for the real symmetric matrix, especially the determination of the largest eigenvalue. The largest eigenvalue is the maximum extremum of the objective function called the Rayleigh quotient and can be determined by the steepest descent method. It is known, however, that the steepest descent method suffers from slow convergence because it converges linearly. Especially, when the largest and the nest largest eigenvalues have very close values, the convergence is particularly slow. This paper analyzes this situation and shows that the convergence can be accelerated by combining the steepest descent method with a technique called shaking. Finally, it is demonstrated by a numerical example that the convergence is accelerated drastically by the proposed technique. [ABSTRACT FROM AUTHOR]

Let R be any ring with identity. An element a ∊ R is called nil-clean, if a = e + n where e is an idempotent element and n is a nil-potent element. In this paper we give necessary and sufficient conditions for a 2 × 2 matrix over an integral domain R to be nil-clean. [ABSTRACT FROM AUTHOR]

The paper investigates classes of words equivalent under different ways of counting subwords. We present a general method of constructing sequences of equivalent words. Apart from obtaining numerical characterizations of words, we also construct sequences of words generating equal arithmetical power sums. Our notions of equivalence include the Parikh equivalence resulting from Parikh matrices, recently widely studied. Consequently, we are able to settle some open problems in this area. [ABSTRACT FROM AUTHOR]

This paper gives a quantitative result that if A is a substochastic matrix and has r eigenvalues which are sufficiently close to 1, then A has r disjoint principal submatrices which are nearly stochastic. [ABSTRACT FROM AUTHOR]

Families of matrices smoothly depending on a vector of parameters are considered. Arnold [Russian Math. Surveys, 26 (1971), pp. 29–43] and Galin [Uspekhi Mat. Nauk, 27 (1972),pp. 241–242] have found and listed normal forms of families of complex and real matrices (miniversal deformations), to which any family of matrices can be transformed in the vicinity of a point in the parameter space by a change of basis, smoothly dependent on a vector of parameters, and by a smooth change of parameters. In this paper a constructive method of determining functions describing a change of basis and a change of parameters, transforming an arbitrary family to the miniversal deformation, is suggested. Derivatives of these functions with respect to parameters are determined from a recurrent procedure using derivatives of the functions of lower orders and derivatives of the family of matrices. Then the functions are found as Taylor series. Examples are given. The suggested method allows using efficiently miniversal deformations for investigation of different properties of matrix families. This is shown in the paper where tangent cones (linear approximations) to the stability domain at the singular boundary points are found. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we apply a complete pivoting strategy to compute the left-looking version of AINV preconditioner for linear systems. As the preprocessing, the MultiLevel Nested Dissection reordering has also been applied. We have used this preconditioner as the right preconditioner for several linear systems where the coefficient matrices have been downloaded from the University of Florida Sparse Matrix Collection. Numerical experiments presented in this paper indicate the effectiveness of such a complete pivoting on the quality of left-looking version of AINV preconditioner. [Copyright &y& Elsevier]

The paper investigates the inclusion intervals of matrix singular values. By employing the digraph of a matrix, some new inclusion intervals of matrix singular values are presented. These intervals are based mainly on the use of positive scale vectors and their intersections. Theoretic analysis and numerical examples show that these results improve and generalize some known results of Li et al. (Numer. Linear Algebra Appl. 2007; 14(2):115–128) and Kolotilina (J. Math. Sci. 2006; 137(3):4794–4800). Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Scaling is a commonly used technique for standard eigenvalue problems to improve the sensitivity of the eigenvalues. In this paper we investigate scaling for generalized and polynomial eigenvalue problems (PEPs) of arbitrary degree. It is shown that an optimal diagonal scaling of a PEP with respect to an eigenvalue can be described by the ratio of its normwise and componentwise condition number. Furthermore, the effect of linearization on optimally scaled polynomials is investigated. We introduce a generalization of the diagonal scaling by Lemonnier and Van Dooren to PEPs that is especially effective if some information about the magnitude of the wanted eigenvalues is available and also discuss variable transformations of the type λ = αμ for PEPs of arbitrary degree. [ABSTRACT FROM AUTHOR]

In this paper, based on Patel's algorithm (1993), we propose a structure-preserving algorithm for solving palindromic quadratic eigenvalue problems (QEPs). We also show the relationship between the structure-preserving algorithm and the URV-based structure-preserving algorithm by Schröder (2007). For large sparse palindromic QEPs, we develop a generalized T-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi algorithm for solving the resulting T-skew-Hamiltonian pencils. Numerical experiments show that our proposed structure-preserving algorithms perform well on the palindromic QEP arising from a finite element model of high-speed trains and rails. [ABSTRACT FROM AUTHOR]

Abstract: Recently, Shen et al. has published a paper concerning the properties and computations of matrix pseudospectra. However, some of the main results seem to be incomplete. As a complement of the paper, in this note we first make several examples to show the incompleteness of that results, then we give the modified results through giving more appropriate definitions replacing the one given in [Y. Shen, J. Zhao, H. Fan, Properties and computations of matrix pseudospectra, Applied Mathematics and Computation 161 (2005) 385–393]. An interesting inclusion theorem for pseudospectra is given, which provides preferable bounds for standard pseudospectra. Some other properties are also explored. [Copyright &y& Elsevier]

This paper presents an approach to transform asymmetric systems into symmetric systems by equivalence transformation and discusses what forms of restrictions should be imposed on the system matrices so that they can be simultaneously transformed into symmetric matrices. Conditions of symmetrizability obtained here are more "liberal" and numerical calculations of this transformation are more straightforward. Several examples are provided to illustrate the new approach. [ABSTRACT FROM AUTHOR]

This paper suggests a new method, called AINV-A, for constructing sparse approximate inverse preconditioners for positive-definite matrices, which can be regarded as a modification of the AINV method proposed by Benzi and Túma. Numerical results on SPD test matrices coming from different applications demonstrate the robustness of the AINV-A method and its superiority to the original AINV approach. Copyright © 2001 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the iterative solution of ill-conditioned Toeplitz systems generated by continuous nonnegative real-valued functions f with a finite number of zeros. We construct new w-circulant preconditioners without explicit knowledge of the generating function f by approximating f by its convolution f * KN with a suitable positive reproducing kernel KN. By the restriction to positive kernels we obtain positive definite preconditioners. Moreover, if f has only zeros of even order ≤ 2s, then we can prove that the property [This symbol cannot be presented in ASCII format]π-π t2k KN(t)dt ≤ CN-2k (k = 0, … ,s) of the kernel is necessary and sufficient to ensure the convergence of the PCG method in a number of iteration steps independent of the dimension N of the system. Our theoretical results were confirmed by numerical tests. [ABSTRACT FROM AUTHOR]

This paper extends the author's earlier work which proved that, for a certain class of Hermitian eigenproblems AP = λBP (including, most notably, the Toeplitz eigenproblem), the eigenvectors can be represented as saddlepoints of a special form of the Rayleigh quotient. Each saddlepoint solves a min-max/max-min optimization problem whose optimal value is the eigenvalue. The zeros of the eigenpolynomial factors are located with respect to the unit circle. These results were proved, in part, by using an inertia theorem for Stein equations. For another class of Hermitian eigenproblems AP = λBP, an analogous set of results are derived here using an inertia theorem for Lyapunov equations. In this case, the zeros of the eigenpolynomial factors are located with respect to the imaginary axis, and sufficient conditions for the eigenpolynomial factors to have only extended imaginary zeros are established. The highlight of the article is an important theorem that yields a general factorization of saddlepoint eigenpolynomials and parameterizes all eigenvector saddlepoint representations for an m-dimensional eigenspace. Several key extensions to the theory are also presented. [ABSTRACT FROM AUTHOR]

In this paper we exhibit, under suitable conditions, a neat relationship between the Moore­Penrose generalized inverse of a sum of two matrices and the Moore­Penrose generalized inverses of the individual terms. We include an application to the parallel sum of matrices. [ABSTRACT FROM AUTHOR]

In this paper we provide an actual bound for the distance between the original and the perturbed right singular vector subspaces of a general matrix with full column rank. We also provide actual relative componentwise bounds for perturbed eigenvectors of a positive definite matrix. [ABSTRACT FROM AUTHOR]

Superlinear preconditioners (those that provide a proper cluster at 1) are very important for the cg-like methods since they make these methods converge superlinearly. As is well known, for Toeplitz matrices generated by a continuous symbol, many circulant and circulant-like (related to different matrix algebras) preconditioners were proved to be superlinear. In contrast, for multilevel Toeplitz matrices there has been no proof of the superlinearity of any multilevel circulants. In this paper we show that such a proof is not possible since any multilevel circulant preconditioner is not superlinear, in the general case of multilevel Toeplitz matrices. Moreover, for matrices not necessarily Toeplitz, we present some general results proving that many popular structured preconditioners cannot be superlinear. [ABSTRACT FROM AUTHOR]

Let WN = WN(z1,z2,…,zn) be a rectangular Vandermonde matrix of order n×N, N ≥ n, with distinct nodes zj in the unit disk and zjk-1 as its (j,k) entry. Matrices of this type often arise in frequency estimation and system identification problems. In this paper, the conditioning of WN is analyzed and bounds for the spectral condition numberk2(WN) are derived. The bounds depend on n, N, and the separation of the nodes. By analyzing the behavior of the bounds as functions of N, we conclude that these matrices may become well conditioned, provided the nodes are close to the unit circle but not extremely close to each other and provided the number of columns of WN is large enough. The asymptotic behavior of both the conditioning itself and the bounds is analyzed and the theoretical results arising from this analysis verified by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper we consider real or complex skew-Hamiltonian/Hamiltonian pencils λS - H, i.e., pencils where S is a skew-Hamiltonian and H is a Hamiltonian matrix. These pencils occur, for example, in the theory of continuous time, linear quadratic optimal control problems. We reduce these pencils to canonical and Schur-type forms under structure-preserving transformations, i.e., J-congruence-transformations (λS-H) → -JP*J(λS-H)P, where P is nonsingular or unitary. [ABSTRACT FROM AUTHOR]

We present a fast condition estimation algorithm for the eigenvalues of a class of structured matrices. These matrices are low rank modifications to Hermitian, skew-Hermitian, and unitary matrices. Fast structured operations for these matrices are presented, including Schur decomposition, eigenvalue block swapping, matrix equation solving, compact structure reconstruction, etc. Compact semiseparable representations of matrices are used in these operations. We use these operations in a new, improved version of the statistical condition estimation method for eigenvalue problems. The estimation algorithm costs O(n²) flops for all eigenvalues, instead of O(n³) as in traditional algorithms, where n is the order of the matrix. The algorithm provides reliable condition estimates for both eigenvalues and eigenvalue clusters. The proposed structured matrix operations are also useful for additional eigenvalue problems and other applications. Numerical examples are used to illustrate the reliability and efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

It is well known that there exists a transversal design TDλ[ k; u] admitting a class regular automorphism group U if and only if there exists a generalized Hadamard matrix GH( u, λ) over U. Note that in this case the resulting transversal design is symmetric by Jungnickel’s result. In this article we define a modified generalized Hadamard matrix and show that transversal designs which are not necessarily symmetric can be constructed from these under a modified condition similar to class regularity even if it admits no class regular automorphism group. [ABSTRACT FROM AUTHOR]

We study the matrix equation C(BXC)†B = X†, where X† denotes the Moore-Penrose inverse. We derive conditions for the consistency of the equation and express all its solutions using singular vectors of B and C. Applications to compliance matrices in molecular dynamics, to mixed reverse-order laws of generalized inverses and to weighted Moore-Penrose inverses are given. [ABSTRACT FROM AUTHOR]

We consider DG-methods for second order scalar elliptic problems using piecewise affine approximation in two or three space dimensions. We prove that both the symmetric and the nonsymmetric versions of the DG-method have regular system matrices without penalization of the interelement solution jumps provided boundary conditions are imposed in a certain weak manner. Optimal convergence is proved for sufficiently regular meshes and data. We then propose a DG-method using piecewise affine functions enriched with quadratic bubbles. Using this space we prove optimal convergence in the energy norm for both a symmetric and nonsymmetric DG-method without stabilization. All of these proposed methods share the feature that they conserve mass locally independent of the penalty parameter. [ABSTRACT FROM AUTHOR]

We present computationally efficient criteria that can guarantee correct ordinal ranking of Google's PageRank scores when they are computed with the power method (ordinal ranking of a list consists of assigning an ordinal number to each item in the list). We discuss the tightness of the ranking criteria, and illustrate their effectiveness for top k and bucket ranking. We present a careful implementation of the power method, combined with a roundoff error analysis that is valid for matrix dimensions n < 1014. To first order, the roundoff error depends neither on n nor on the iteration count, but only on the maximal number of inlinks and the dangling nodes. The applicability of our ranking criterion is limited by the roundoff error from a single matrix vector multiply. Numerical experiments suggest that our criteria can effectively rank the top PageRank scores. We also discuss how to implement ranking for extremely large practical problems, by curbing roundoff error, reducing the matrix dimension, and using faster converging methods [ABSTRACT FROM AUTHOR]

We propose a fast verification method for saddle point linear systems where the (1,1) block is singular. The proposed verification method is based on an algebraic analysis of a block diagonal preconditioner and rounding mode controlled computations. Numerical comparison of several verification methods with various block diagonal preconditioners is given. [ABSTRACT FROM AUTHOR]

The matrix exponential is a much-studied matrix function having many applications. The Fréchet derivative of the matrix exponential describes the first-order sensitivity of eA to perturbations in A and its norm determines a condition number for eA. Among the numerous methods for computing eA the scaling and squaring method is the most widely used. We show that the implementation of the method in [N. J. Higham, The scaling and squaring method for the matrix exponential revisited, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 1179-1193] can be extended to compute both eA and the Fréchet derivative at A in the direction E, denoted by L(A,E), at a cost about three times that for computing eA alone. The algorithm is derived from the scaling and squaring method by differentiating the Padé approximants and the squaring recurrence, reusing quantities computed during the evaluation of the Padé approximant, and intertwining the recurrences in the squaring phase. To guide the choice of algorithmic parameters, an extension of the existing backward error analysis for the scaling and squaring method is developed which shows that, modulo rounding errors, the approximations obtained are eA+ΔA and L(A + ΔA,E + ΔE), with the same ΔA in both cases, and with computable bounds on ∥ ΔA ∥ and ∥ΔE ∥. The algorithm for L(A,E) is used to develop an algorithm that computes eA together with an estimate of its condition number. In addition to results specific to the exponential, we develop some results and techniques for arbitrary functions. We show how a matrix iteration for f(A) yields an iteration for the Fréchet derivative and show how to efficiently compute the Fréchet derivative of a power series. We also show that a matrix polynomial and its Fréchet derivative can be evaluated at a cost at most three times that of computing the polynomial itself and give a general framework for evaluating a matrix function and its Fréchet derivative via Padé approximation. [ABSTRACT FROM AUTHOR]

Computing the Candecomp/Parafac (CP) solution of R components (i.e., the best rank-R approximation) for a generic IxJx2 array may result in diverging components, also known as "degeneracy." In such a case, several components are highly correlated in all three modes, and their component weights become arbitrarily large. Evidence exists that this is caused by the nonexistence of an optimal CP solution. Instead of using CP, we propose to compute the best approximation by means of a generalized Schur decomposition (GSD), which always exists. The obtained GSD solution is the limit point of the sequence of CP updates (whether it features diverging components or not) and can be separated into a nondiverging CP part and a sparse Tucker3 part or into a nondiverging CP part and a smaller GSD part. We show how to obtain both representations and illustrate our results with numerical experiments. [ABSTRACT FROM AUTHOR]

Hyperbolic quadratic matrix polynomials Q(λ) = λ²A + λB + C are an important class of Hermitian matrix polynomials with real eigenvalues, among which the overdamped quadratics are those with nonpositive eigenvalues. Neither the definition of overdamped nor any of the standard characterizations provides an efficient way to test if a given Q has this property. We show that a quadratically convergent matrix iteration based on cyclic reduction, previously studied by Guo and Lancaster, provides necessary and sufficient conditions for Q to be overdamped. For weakly overdamped Q the iteration is shown to be generically linearly convergent with constant at worst 1/2, which implies that the convergence of the iteration is reasonably fast in almost all cases of practical interest. We show that the matrix iteration can be implemented in such a way that when overdamping is detected a scalar μ < 0 is provided that lies in the gap between the n largest and n smallest eigenvalues of the n x n quadratic eigenvalue problem (QEP) Q(λ)x = 0. Once such a μ is known, the QEP can be solved by linearizing to a definite pencil that can be reduced, using already available Cholesky factorizations, to a standard Hermitian eigenproblem. By incorporating an initial preprocessing stage that shifts a hyperbolic Q so that it is overdamped, we obtain an efficient algorithm that identifies and solves a hyperbolic or overdamped QEP maintaining symmetry throughout and guaranteeing real computed eigenvalues [ABSTRACT FROM AUTHOR]

In terms of the generalized singular value decomposition, we define the generalized singular matrix sets and their backward errors. The explicit expressions of backward errors are derived, which extend a result of Sun [SIAM J. Matrix Anal. Appl., 22 (2000), pp. 323-341]. The results are illustrated by a numerical example. [ABSTRACT FROM AUTHOR]

The article offers information on a strong version of Liouville's theorem. This theorem states that every bounded holomorphic function on C is constant. The authors, through various mathematical problems and their corresponding solutions, present an elementary proof of this theorem. They were also able to obtain a corresponding statement for complex-value functions by applying Theorem 1.1. Moreover, different mathematical propositions and their corresponding numerical analysis and solutions are presented.

We present a compact discontinuous Galerkin (CDG) method for an elliptic model problem. The problem is first cast as a system of first order equations by introducing the gradient of the primal unknown, or flux, as an additional variable. A standard discontinuous Galerkin (DG) method is then applied to the resulting system of equations. The numerical interelement fluxes are such that the equations for the additional variable can be eliminated at the element level, thus resulting in a global system that involves only the original unknown variable. The proposed method is closely related to the local discontinuous Galerkin (LDG) method [B. Cockburn and C.-W. Shu, SIAM J. Numer. Anal., 35 (1998), pp. 2440-2463], but, unlike the LDG method, the sparsity pattern of the CDG method involves only nearest neighbors. Also, unlike the LDG method, the CDG method works without stabilization for an arbitrary orientation of the element interfaces. The computation of the numerical interface fluxes for the CDG method is slightly more involved than for the LDG method, but this additional complication is clearly offset by increased compactness and flexibility. Compared to the BR2 [F. Bassi and S. Rebay, J. Comput. Phys., 131 (1997), pp. 267-279] and IP [J. Douglas, Jr., and T. Dupont, in Computing Methods in Applied Sciences (Second Internat. Sympos., Versailles, 1975), Lecture Notes in Phys. 58, Springer, Berlin, 1976, pp. 207-216] methods, which are known to be compact, the present method produces fewer nonzero elements in the matrix and is computationally more efficient. [ABSTRACT FROM AUTHOR]

Mixed finite element formulations give rise to large, sparse, block linear systems of equations, the solution of which is often sought via a preconditioned iterative technique. In this work we present a general analysis of block-preconditioners based on the stability conditions inherited from the formulation of the finite element method (the Babuška-Brezzi, or inf-sup, conditions). The analysis is motivated by the notions of norm-equivalence and field-of-values-equivalence of matrices. In particular, we give sufficient conditions for diagonal and triangular block-preconditioners to be norm- and field-of-values-equivalent to the system matrix. [ABSTRACT FROM AUTHOR]

A class ξ of algebras of symmetric n × n matrices, related to Toeplitz-plus-Hankel structures and including the well-known algebra H diagonalized by the Hartley transform, is investigated. The algebras of ξ are then exploited in a general displacement decomposition of an arbitrary n × n matrix A. Any algebra of ξ is a 1-space, i.e., it is spanned by n matrices having as first rows the vectors of the canonical basis. The notion of 1-space (which generalizes the previous notions of L1 space [Bevilacqua and Zellini, Linear and Multilinear Algebra, 25 (1989), pp. 1–25] and Hessenberg algebra [Di Fiore and Zellini, Linear Algebra Appl., 229 (1995), pp. 49–99]) finally leads to the identification in ξ of three new (non-Hessenberg) matrix algebras close to H, which are shown to be associated with fast Hartley-type transforms. These algebras are also involved in new efficient centrosymmetric Toeplitz-plus-Hankel inversion formulas. [ABSTRACT FROM AUTHOR]

For nonsingular n×n matrices A, φ(A) = AοA-T(ο denotes the Hadamard product and A-T the inverse transpose, (A-1)T , of A) arises in mathematical control theory associated with chemical engineering design problems and in a matrix theoretic problem involving the relation between the diagonal entries and eigenvalues. For any positive integer t, we show that the equation [This symbol cannot be presented in ASCII format] has at least 2t-1and 2t mutually φ-distinct real solutions for the case n = 4 and n > 4, respectively, and that these solutions can be determined by an inverted iteration. We also prove that the above equation has only one φ-distinct real solution (1¹ -1¹) for t = 1 and has no real solutions for t ≥ 2 when n = 2. The above results answer the problem that has been proposed by Johnson and Shapiro in [SIAM J. Algebraic Discrete Methods, 7 (1986), pp. 627–644]. When does the equation [This symbol cannot be presented in ASCII format] have a real solution? [ABSTRACT FROM AUTHOR]

The second logarithmic derivative μ∝(2) [A] of a complex n × n matrix A in the Chebyshev norm is defined as the second right derivative of [This symbol cannot be presented in ASCII format], where . ·. ∝ denotes the operator norm corresponding to the norm . ·. ∝ in Cn. The obtained formula is illustrated by a numerical example. The result may be of interest in applications such as stability theory. [ABSTRACT FROM AUTHOR]

We present new O(n3) algorithms to compute very accurate singular value decompositions of Cauchy matrices, Vandermonde matrices, and related "unit-displacement-rank" matrices. These algorithms compute all the singular values with guaranteed relative accuracy, independent of their dynamic range. In contrast, previous O(n3) algorithms can potentially lose all relative accuracy in the tiniest singular values. [ABSTRACT FROM AUTHOR]

The linear complementarity problem (q,A) with data A ∈ Rn×n and q ∈ Rn involves finding a nonnegative z ∈ Rn such that Az +q ≥ 0 and zt(Az +q) = 0. Cottle and Stone introduced the class of P1-matrices and showed that if A is in P1\Q, then K(A) (the set of all q for which (q,A) has a solution) is a half-space and (q,A) has a unique solution for every q in the interior of K(A). Extending the results of Murthy, Parthasarathy, and Sriparna [Ann. Dynamic Games, to appear], we present a number of equivalent characterizations of P1\Q. Also, we present yet another characterization of P-matrices. This widens the range of matrix classes for which a conjecture raised by Murthy, Parthasarathy, and Sriparna [SIAM J. Matrix Anal. Appl., 19 (1998), pp. 898­905] characterizing the class of Lipschitzian matrices is true. [ABSTRACT FROM AUTHOR]

A method for finding the best approximation of a matrix A by a full rank Hankel matrix is given. The initial problem of best approximation of one matrix by another is transformed to a problem involving best approximation of a given vector by a second vector whose elements are constrained so that its inverse image is a Hankel matrix. The map from a matrix to a vector is the invertible map between a subspace represented as the row space of the matrix A and the Grassman vector representing that subspace. The relation between the principle angles associated with a pair of subspaces and the angle between the Grassman vectors associated with the subspaces is established. [ABSTRACT FROM AUTHOR]

We derive upper and lower bounds on the smallest and largest eigenvalues, respectively, of real symmetric Toeplitz matrices. The bounds are first obtained for positive-definite matrices and then extended to the general real symmetric case. They are computed as the roots of rational and polynomial approximations to spectral, or secular, equations for the symmetric and antisymmetric parts of the spectrum; this leads to separate bounds on the even and odd eigenvalues. We also present numerical results. [ABSTRACT FROM AUTHOR]

Let T(bjk) be a finite collection of infinite Toeplitz band matrices, let Tn(bjk) denote their (n + 1) × (n + 1) truncations, and put [This symbol cannot be presented in ASCII format] where . ·. p stands for the operator norm associated with the lp-norm 1 ≤ p < ∝ ) on Cn+1. We establish tight two-sided estimates for the difference Mp - . An. p. It is well known that if T(b) is a Hermitian Toeplitz band matrix and An = Tn(b), then M2 - . An. 2 goes to zero with polynomial speed. We show that such a slow convergence rate is, in a sense, an exceptional case, and we prove that in the generic case Mp - . An. p approaches zero with exponential speed. Our results yield good error estimates when computing the norms of certain infinite matrices via their large truncations and, conversely, when determining the norms of certain large matrices by having recourse to known norms of infinite matrices. [ABSTRACT FROM AUTHOR]

Let A = (A1,…,Am) be an m-tuple of n × n Hermitian matrices. For 1 ≤ k ≤ n, the kth joint numerical range of A is defined by [This symbol cannot be presented in ASCII format] We consider linearly independent families of Hermitian matrices {A1,…,Am} so that Wk(A) is convex. It is shown that m can reach the upper bound 2k(n - k) + 1. A key idea in our study is relating the convexity of Wk(A) to the problem of constructing rank k orthogonal projections under linear constraints determined by A. The techniques are extended to study the convexity of other generalized numerical ranges and the corresponding matrix construction problems. [ABSTRACT FROM AUTHOR]