Linear operators preserving the decomposable numerical range

Multilinear techniques are used to characterize unitary matrices in terms of a generalized numerical range. This characterization is then applied to analyze the structure of all linear operators on matrices which preserve this numerical range. The results generalize V. J. Pellegrini's determination of all linear operators preserving the classical numerical range.     

A gentle guide to the basics of two projections theory

This paper is a survey of the basics of the theory of two projections. It contains in particular the theorem by Halmos on two orthogonal projections and Roch, Silbermann, Gohberg, and Krupnik’s theorem on two idempotents in Banach algebras. These two theorems, which deliver the desired results usually very quickly and comfortably, are missing or wrongly cited in many recent publications on the topic, The paper is intended as a gentle guide to the field. The basic theorems are precisely stated, some of them are accompanied by full proofs, others not, but precise references are given in each case, and many examples illustrate how to work with the theorems.     

Dimension inequalities for the closure of the image of a multilinear function

In this note we give a simple proof and an extension of a dimension inequality of Howard concerning the range of a multilinear function with vector space range by using some results on algebraic varieties.     

Max algebraic powers of irreducible matrices in the periodic regime: An application of cyclic classes

In max algebra it is well known that the sequence of max algebraic powers A^k, with A an irreducible square matrix, becomes periodic after a finite transient time T(A), and the ultimate period γ is equal to the cyclicity of the critical graph of A.In this connection, we study computational complexity of the following problems: (1) for a given k, compute a periodic power A^r with and r?T(A), (2) for a given x, find the ultimate period of {A^l⊗x}. We show that both problems can be solved by matrix squaring in O(n³logn) operations. The main idea is to apply an appropriate diagonal similarity scaling A?X^-1AX, called visualization scaling, and to study the role of cyclic classes of the critical graph.     

Singular value estimates of oblique projections

Let W and M be two finite dimensional subspaces of a Hilbert space H such that H=W⊕M^⊥, and let P_W‖M^⊥ denote the oblique projection with range W and nullspace M^⊥. In this article we get the following formula for the singular values of P_W‖M^⊥

     

Full ordering in the Shorrocks mobility sense of the semiring of monotone doubly stochastic matrices

In this paper, we investigate the ordering on a semiring of monotone doubly stochastic transition matrices in Shorrocks’ sense. We identify a class of an equilibrium index of mobility that induces the full ordering in a semiring, while this ordering is compatible with Dardanoni’s partial ordering on a set of monotone primitive irreducible doubly stochastic matrices.     

Semigroup of matrices acting on the max-plus projective space

We investigate the action of semigroups of d×d matrices with entries in the max-plus semifield on the max-plus projective space. Recall that semigroups generated by one element with projectively bounded image are projectively finite and thus contain idempotent elements.In terms of orbits, our main result states that the image of a minimal orbit by an idempotent element of the semigroup with minimal rank has at most d! elements. Moreover, each idempotent element with minimal rank maps at least one orbit onto a singleton.This allows us to deduce the central limit theorem for stochastic recurrent sequences driven by independent random matrices that take countably many values, as soon as the semigroup generated by the values contains an element with projectively bounded image.     

A concise proof of Kruskal’s theorem on tensor decomposition

A theorem of J. Kruskal from 1977, motivated by a latent-class statistical model, established that under certain explicit conditions the expression of a third-order tensor as the sum of rank-1 tensors is essentially unique. We give a new proof of this fundamental result, which is substantially shorter than both the original one and recent versions along the original lines.     

Quivers of finite mutation type and skew-symmetric matrices

Quivers of finite mutation type are certain directed graphs that first arised in Fomin-Zelevinsky’s theory of cluster algebras. It has been observed that these quivers are also closely related with different areas of mathematics. In fact, main examples of finite mutation type quivers are the quivers associated with triangulations of surfaces. In this paper, we study structural properties of finite mutation type quivers in relation with the corresponding skew-symmetric matrices. We obtain a characterization of finite mutation type quivers that are associated with triangulations of surfaces and give a new numerical invariant for their mutation classes.     

Geometric properties of positive definite matrices cone with respect to the Thompson metric

We shall discuss geometric properties of a quadrangle with parallelogramic properties in a convex cone of positive definite matrices with respect to Thompson metric.     

Boundary Behavior of Functions in the de Branges–Rovnyak Spaces

This paper deals with the boundary behavior of functions in the de Branges–Rovnyak spaces. First, we give a criterion for the existence of radial limits for the derivatives of functions in the de Branges–Rovnyak spaces. This criterion generalizes a result of Ahern–Clark. Then we prove that the continuity of all functions in a de Branges–Rovnyak space on an open arc I of the boundary is enough to ensure the analyticity of these functions on I. We use this property in a question related to Bernstein’s inequality. Received: May 10, 2007. Revised: August 8, 2007. Accepted: August 8, 2007.     

The product of operators with closed range in Hilbert C-modules

Suppose T and S are bounded adjointable operators with close range between Hilbert C^∗-modules, then TS has closed range if and only if Ker(T)+Ran(S) is an orthogonal summand, if and only if Ker(S^∗)+Ran(T^∗) is an orthogonal summand. Moreover, if the Dixmier (or minimal) angle between Ran(S) and Ker(T)∩[Ker(T)∩Ran(S)]^⊥ is positive and is an orthogonal summand then TS has closed range.     

An a posteriori error estimation of the Rayleigh-Ritz eigenvalues

Summary LetA, B be essentially self-adjoint and positive definite differential operators defined inL ₂(G). Using Svirskij's construction of the base operator and some results from the analytic perturbation theory of linear operators a formula providing eigenvalue lower bounds of the problemAu=Bu is derived. In this formula a rough lower bound of some higher eigenvalue and the residual convergence of the Rayleigh-Ritz eigenfunction approximations are needed. Some numerical results are presented.     

Extremal patterns of distinct entries in vectors in the range of a matrix

For an m × n matrix A over a field F we consider the following quantities: μ(A), the maximum multiplicity of a field element as a component of a nonzero vector in the range of A, and δ(A), the minimum number of distinct entries in a nonzero vector in the range of A. In terms of ramk(A), we describe the set of possible values of μand δ and discuss the possible relations between them. We also develop a general affine geometric structure in which the sets of values of μ and δ may be characterized linear algebraically.     

Linear algorithms for testing the sign stability of a matrix and for findingZ-maximum matchings in acyclic graphs

Summary Ann×n real matrixA=(a _ij ) isstable if each eigenvalue has negative real part, andsign stable (orqualitatively stable) if each matrix B with the same sign-pattern asA is stable, regardless of the magnitudes ofB's entries. Sign stability is of special interest whenA is associated with certain models from ecology or economics in which the actual magnitudes of thea _ij may be very difficult to determine. Using a characterization due to Quirk and Ruppert, and to Jeffries, an efficient algorithm is developed for testing the sign stability ofA. Its time-and-space-complexity are both 0(n ²), and whenA is properly presented that is reduced to 0(max{n, number of nonzero entries ofA}). Part of the algorithm involves maximum matchings, and that subject is treated for its own sake in two final sections.     

Minimal rank of abelian group matrices

The minimal rank of abelian group matrices with positive integral entries is determined.The corresponding problem for circulant matrices have been investigated by Ingleton and more recently by Shiu-Ma-Fang. Our work can be viewed as a generalization of their results, since a group matrix becomes circulant when the group is cyclic.     

The convergence of a direct BEM for the plane mixed boundary value problem of the Laplacian

Summary In this paper the convergence analysis of a direct boundary elecment method for the mixed boundary value problem for Laplace equation in a smooth plane domain is given. The method under consideration is based on the collocation solution by constant elements of the corresponding system of boundary integral equations. We prove the convergence of this method, provide asymptotic error estimates for the BEM-solution and give some numerical examples.     

Vandermonde matrices on the circle: Spectral properties and conditioning

Summary We study Vandermonde matrices whose nodes are given by a Van der Corput sequence on the unit circle. Our primary interest is in the singular values of these matrices and the respective (spectral) condition numbers. Detailed information about multiplicities and eigenvectors, however, is also obtained. Two applications are given to the theory of polynomials.Dedicated to R. S. Varga on the occasion of his sixtieth birthdayResearch of A. C. supported by the Fundación Andes, Chile, and by the German Academic Exchange Service (DAAD), Federal Republic of GermanyResearch of W. G. supported, in part, by the National Science Foundation, USA, (Grant CCR-8704404)Research of S. R. supported by the Fondo Nacional de Desarollo Cientßfico y Tecnológico (FONDECYT), Chile, (Grant 237/89), by the Universidad Técnica F. Santa Marßa, Valparaßso, Chile, (Grant 89.12.06), and by the German Academic Exchange Service (DAAD), Federal Republic of Germany     

On condition numbers and the distance to the nearest ill-posed problem

Summary The condition number of a problem measures the sensitivity of the answer to small changes in the input. We call the problem ill-posed if its condition number is infinite. It turns out that for many problems of numerical analysis, there is a simple relationship between the condition number of a problem and the shortest distance from that problem to an ill-posed one: the shortest distance is proportional to the reciprocal of the condition number (or bounded by the reciprocal of the condition number). This is true for matrix inversion, computing eigenvalues and eigenvectors, finding zeros of polynomials, and pole assignment in linear control systems. In this paper we explain this phenomenon by showing that in all these cases, the condition number satisfies one or both of the diffrential inequalitiesm·²DM·², where D is the norm of the gradient of . The lower bound on D leads to an upper bound 1/m(x) on the distance. fromx to the nearest ill-posed problem, and the upper bound on D leads to a lower bound 1/(M(X)) on the distance. The attraction of this approach is that it uses local information (the gradient of a condition number) to answer a global question: how far away is the nearest ill-posed problem? The above differential inequalities also have a simple interpretation: they imply that computing the condition number of a problem is approximately as hard as computing the solution of the problem itself. In addition to deriving many of the best known bounds for matrix inversion, eigendecompositions and polynomial zero finding, we derive new bounds on the distance to the nearest polynomial with multiple zeros and a new perturbation result on pole assignment.     

The study of Jacobi and cyclic Jacobi matrix eigenvalue problems using Sturm-Liouville theory

We study the eigenvalues of matrix problems involving Jacobi and cyclic Jacobi matrices as functions of certain entries. Of particular interest are the limits of the eigenvalues as these entries approach infinity. Our approach is to use the recently discovered equivalence between these problems and a class of Sturm-Liouville problems and then to apply the Sturm-Liouville theory.