Linear operators preserving the numerical range (radius) on triangular matrices

We characterize those linear operators on triangular or diagonal matrices preserving the numerical range or radius.     

Local convergence of the (exact and inexact) iterative aggregation method for linear systems and Markov operators

Summary. The iterative aggregation method for the solution of linear systems is extended in several directions: to operators on Banach spaces; to the method with inexact correction, i.e., to methods where the (inner) linear system is in turn solved iteratively; and to the problem of finding stationary distributions of Markov operators. Local convergence is shown in all cases. Convergence results apply to the particular case of stochastic matrices. Moreover, an argument is given which suggests why the iterative aggregation method works so well for nearly uncoupled Markov chains, as well as for Markov chains with other zero-nonzero structures. Received May 25, 1991/Revised version received February 23, 1994     

Bijective linear rank preservers for spaces of matrices over antinegative semirings

We classify the bijective linear operators on spaces of matrices over antinegative commutative semirings with no zero divisors which preserve certain rank functions such as the symmetric rank, the factor rank and the tropical rank. We also classify the bijective linear operators on spaces of matrices over the max-plus semiring which preserve the Gondran-Minoux row rank or the Gondran-Minoux column rank.     

Linear operators on matrices: the invariance of the decomposable numerical range

The purpose of this note is to study the structure of all linear operators on matrices which preserve the generalized numerical range. The result generalizes V. J. Pellegrini's determination of all linear operators preserving the classical numerical range in matrix version.     

Fischer determinantal inequalities and Highamʼs Conjecture

We obtain a Fischer type determinantal inequality for matrices with given angular numerical range. We discuss the growth factor for Gaussian elimination for linear systems in which the coefficient matrix has this form and give a proof of Higham?s Conjecture.     

Product numerical range in a space with tensor product structure

We study operators acting on a tensor product Hilbert space and investigate their product numerical range, product numerical radius and separable numerical range. Concrete bounds for the product numerical range for Hermitian operators are derived. Product numerical range of a non-Hermitian operator forms a subset of the standard numerical range containing the barycenter of the spectrum. While the latter set is convex, the product range needs not to be convex nor simply connected. The product numerical range of a tensor product is equal to the Minkowski product of numerical ranges of individual factors.     

Zero-term rank preservers

We obtain characterizations of those linear operators that preserve zero-term rank on the m×n matrices over antinegative semirings. That is, a linear operator T preserves zero-term rank if and only if it has the form T(X)=P(B∘X)Q, where P, Q are permutation matrices and B∘X is the Schur product with B whose entries are all nonzero and not zero-divisors.     

Discriminant preserving linear maps

Consider the n-square matrices over an infiniie field Kas an n²-dimcnsional vector space M( nK). We determine all linear maps Ton M(nK) such that discriminant TX- discriminant Xfor all Xin M(nK)     

Zero-preserving iso-spectral flows based on parallel sums

Driessel [K.R. Driessel, Computing canonical forms using flows, Linear Algebra Appl 379 (2004) 353-379] introduced the notion of quasi-projection onto the range of a linear transformation from one inner product space into another inner product space. Here we introduce the notion of quasi-projection onto the intersection of the ranges of two linear transformations from two inner product spaces into a third inner product space. As an application, we design a new family of iso-spectral flows on the space of symmetric matrices that preserves zero patterns. We discuss the equilibrium points of these flows. We conjecture that these flows generically converge to diagonal matrices. We perform some numerical experiments with these flows which support this conjecture. We also compare our zero-preserving flows with the Toda flow.     

On linear preservers of normal maps

We study group induced cone (GIC) orderings generating normal maps. Examples of normal maps cover, among others, the eigenvalue map on the space of n × n Hermitian matrices as well as the singular value map on n × n complex matrices. In this paper, given two linear spaces equipped with GIC orderings induced by groups of orthogonal operators, we investigate linear operators preserving normal maps of the orderings. A characterization of the preservers is obtained in terms of the groups. The result is applied to show that the normal structure of the spaces is preserved under the action of the operators. In addition, examples are given.     

Schur product of matrices and numerical radius (range) preserving maps

Let F(A) be the numerical range or the numerical radius of a square matrix A. Denote by A ° B the Schur product of two matrices A and B. Characterizations are given for mappings on square matrices satisfying F(A ° B) = F(?(A) ° ?(B)) for all matrices A and B. Analogous results are obtained for mappings on Hermitian matrices.     

$\mathcal{H}$-Matrix approximation for the operator exponential with applications

Summary. We develop a data-sparse and accurate approximation to parabolic solution operators in the case of a rather general elliptic part given by a strongly P-positive operator [4]. In the preceding papers [12]–[17], a class of matrices (-matrices) has been analysed which are data-sparse and allow an approximate matrix arithmetic with almost linear complexity. In particular, the matrix-vector/matrix-matrix product with such matrices as well as the computation of the inverse have linear-logarithmic cost. In the present paper, we apply the -matrix techniques to approximate the exponent of an elliptic operator. Starting with the Dunford-Cauchy representation for the operator exponent, we then discretise the integral by the exponentially convergent quadrature rule involving a short sum of resolvents. The latter are approximated by the -matrices. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different time values. In the case of smooth data (coefficients, boundaries), we prove the linear-logarithmic complexity of the method. Received June 22, 2000 / Revised version received June 6, 2001 / Published online October 17, 2001     

Reverse order laws in C-algebras

In this paper, we offer purely algebraic necessary and sufficient conditions for reverse order laws for generalized inverses in C^∗-algebras, extending rank conditions for matrices and range conditions for Hilbert space operators.     

The condition number of real Vandermonde, Krylov and positive definite Hankel matrices

Summary. We show that the Euclidean condition number of any positive definite Hankel matrix of order may be bounded from below by with , and that this bound may be improved at most by a factor . Similar estimates are given for the class of real Vandermonde matrices, the class of row-scaled real Vandermonde matrices, and the class of Krylov matrices with Hermitian argument. Improved bounds are derived for the case where the abscissae or eigenvalues are included in a given real interval. Our findings confirm that all such matrices – including for instance the famous Hilbert matrix – are ill-conditioned already for “moderate” order. As application, we describe implications of our results for the numerical condition of various tasks in Numerical Analysis such as polynomial and rational i nterpolation at real nodes, determination of real roots of polynomials, computation of coefficients of orthogonal polynomials, or the iterative solution of linear systems of equations. Received December 1, 1997 / Revised version received February 25, 1999 / Published online 16 March 2000     

Linear preservers of matrices of rank-2

Let T be a linear operator on the space of all m×n matrices over any field. we prove that if T maps rank-2 matrices to rank-2 matrices then there exist nonsingular matrices U and V such that either T(X)=UXV for all matrices X, or m=n and T(X)=UX^tV for all matrices X where X^t denotes the transpose of X.     

A matrix lower bound

Four essentially different interpretations of a lower bound for linear operators are shown to be equivalent for matrices (involving inequalities, convex sets, minimax problems, and quotient spaces). Properties stated by von Neumann in a restricted case are satisfied by the lower bound. Applications are made to rank reduction, s-numbers, condition numbers, and pseudospectra. In particular, the matrix lower bound is the distance to the nearest matrix with strictly contained row or column spaces, and it occurs in a condition number formula for any consistent system of linear equations, including those that are underdetermined.     

Characterizations of the polynomial numerical hull of degree k

Six characterizations of the polynomial numerical hull of degree k are established for bounded linear operators on a Hilbert space. It is shown how these characterizations provide a natural distinction between interior and boundary points. One of the characterizations is used to prove that the polynomial numerical hull of any fixed degree k for a Toeplitz matrix whose symbol is piecewise continuous approaches all or most of that of the infinite-dimensional Toeplitz operator, as the matrix size goes to infinity.     

Joint numerical range and its generating hypersurface

In this paper, we study the joint numerical range of m-tuples of Hermitian matrices via their generating hypersurfaces. An example is presented which shows the invalidity of an analogous Kippenhahn theorem for the joint numerical range of three Hermitian matrices.