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Lim and Palfia established the notion of the matrix power means for k positive definite matrices ( k ≥ 3 ) and showed that the matrix power means have the information monotonicity for a unital positive linear mapping. In this note, by virtue of the generalized Kantorovich constant, we show counterparts to the information monotonicity of the matrix power means.

The famous Douglas theorem contains four equivalent conditions on the solvability of the operator equation A X = B associated to Hilbert spaces, which was generalized to the Hilbert C ⁎ -module case by Fang et al. (2009) [5] . In this paper we prove that in the Hilbert C ⁎ -module case, the majorization condition and the range inclusion condition are in fact not equivalent.

We derive an algorithm of optimal complexity which determines whether a given matrix is a Cauchy matrix, and which exactly recovers the Cauchy points defining a Cauchy matrix from the matrix entries. Moreover, we study how to approximate a given matrix by a Cauchy matrix with a particular focus on the recovery of Cauchy points from noisy data. We derive an approximation algorithm of optimal complexity for this task, and prove approximation bounds. Numerical examples illustrate our theoretical results. (C) 2015 Elsevier Inc. All rights reserved.

Inspired by examples of small Hilbert matrices, the author proves a property of symmetric totally positive Cauchy matrices, called AT-property, and consequences for the Hilbert matrix.

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.

The boundary eigenvalue problems for the adjoint of a symmetric relation S in a Hilbert space with finite, not necessarily equal, defect numbers, which are related to the selfadjoint Hilbert space extensions of S are characterized in terms of boundary coefficients and the reproducing kernel Hilbert spaces they induce. (C) 2001 Elsevier Science Inc. All rights reserved.

We propose a new fast O( n 2 ) parallel algorithm for solving Cauchy systems of linear equations. We perform an a priori rounding error analysis and obtain componentwise bounds for the forward, backward and residual errors. These bounds indicate that for the class of totally positive Cauchy matrices the new algorithm is forward and backward stable, producing a remarkably high relative accuracy. In particular, Hilbert linear systems, often considered to be too ill-conditioned to be attacked, can be rapidly solved with high precision. The results indicate a close resemblance between the numerical properties of Cauchy matrices and the much-studied Vandermonde matrices. In fact, our proposed Cauchy solver is an analog of the well-known Bjorck–Pereyra algorithm for Vandermonde systems. As a by-product we obtain favorably backward error bounds for the original Bjorck–Pereyra algorithm. Several computed examples illustrate a connection of high relative accuracy to the concepts of effective well-conditioning and total positivity.

We exhibit a Jacobi matrix T which has simple spectrum and integer entries, and 0 commutes with Hilbert's matrix. As an application we replace the computation of the eigenvectors of Hilbert's matrix (a very ill-conditioned problem) by the computation of the eigenvectors of T (a nicely stable numerical problem).

In this paper, some relationships between the Stirling matrix, the Vandermonde matrix, the Benoulli numbers and the Eulerian numbers are studied from a matrix representation of k!S(n,k) which will be called the factorial Stirling matrix, where S(n,k) are the Stirling numbers of the second kind. As a consequence a number of interesting and useful identities are obtained.

The matrix-valued triangle inequalities of R.C. Thompson [Pacific J. Math. 66 (1976) 285–290] are extended to sequences of matrices with real, complex, or quaternion entries. These new matrix inequalities, in turn, imply a natural formulation of the triangle inequality which is valid in certain Hilbert modules over real or complex semisimple matrix algebras.

Analysis on the set of positive definite matrices is involved in many engineering problems. We develop a differential geometric theory of the set of positive definite matrices by means of a specific class of connections introduced into it. Consequently, various dualistic aspects of the set are elucidated. Next, using the theory, we derive new results and interesting geometrical interpretations for matrix approximation, positive definite matrix completion, and linear matrix inequality problems.

The Bergman-Hilbert matrix is the matrix of the Hilbert-Hankel operator on the Bergman space L2a of the disk. We obtain estimates on its norm and prove the existence of eigenvalues.