Many important problems in applied sciences, mathematics, and engineering can be reduced to matrix problems. Moreover, various applications often introduce a special structure into the corresponding matrices, so that their entries can be described by a certain compact formula. Classic examples include Toeplitz matrices, Hankel matrices, Vandermonde matrices, Cauchy matrices, Pick matrices, Bezoutians, controllability and observability matrices, and others. Exploiting these and the more general structures often allows us to obtain elegant solutions to mathematical problems as well as to design more efficient practical algorithms for a variety of applied engineering problems.

Structured matrices have been under close study for a long time and in quite diverse (and seemingly unrelated) areas, for example, mathematics, computer science, and engineering. Considerable progress has recently been made in all these areas, and especially in studying the relevant numerical and computational issues. In the past few years, a number of practical algorithms blending speed and accuracy have been developed. This significant growth is fully reflected in these volumes, which collect 38 papers devoted to the numerous aspects of the topic.

The collection of the contributions to these volumes offers a flavor of the plethora of different approaches to attack structured matrix problems. The reader will find that the theory of structured matrices is positioned to bridge diverse applications in the sciences and engineering, deep mathematical theories, as well as computational and numerical issues. The presentation fully illustrates the fact that the techniques of engineers, mathematicians, and numerical analysts nicely complement each other, and they all contribute to one unified theory of structured matrices.

The book is published in two volumes. The first contains articles on interpolation, system theory, signal and image processing, control theory, and spectral theory. Articles in the second volume are devoted to fast algorithms, numerical and iterative methods, and various applications.

Readership

Graduate students and research mathematicians interested in linear and multilinear algebra, matrix theory, operator theory, numerical analysis, and systems theory and control.

Table of Contents

• H. Dym -- Structured matrices, reproducing kernels and interpolation
• V. Olshevsky and A. Shokrollahi -- A superfast algorithm for confluent rational tangential interpolation problem via matrix-vector multiplication for confluent Cauchy-like matrices
• S. A. Goreinov and E. E. Tyrtyshnikov -- The maximal-volume concept in approximation by low-rank matrices
• M. H. Gutknecht -- A matrix interpretation of the extended Euclidean algorithm
• V. M. Adukov -- The essential polynomial approach to convergence of matrix Padé approximants

• P. Dewilde -- Systems of low Hankel rank: A survey
• E. Kofidis and P. A. Regalia -- Tensor approximation and signal processing applications
• I. K. Proudler -- Exploiting Toeplitz-like structure in adaptive filtering algorithms using signal flow graphs
• N. Mastronardi, P. Lemmerling, and S. Van Huffel -- The structured total least squares problem
• W. K. Cochran, R. J. Plemmons, and T. C. Torgersen -- Exploiting Toeplitz structure in atmospheric image restoration

• A. C. Antoulas, D. C. Sorensen, and S. Gugercin -- A survey of model reduction methods for large-scale systems
• B. N. Datta and D. R. Sarkissian -- Theory and computations of some inverse Eigenvalue problems for the quadratic pencil
• D. Calvetti, B. Lewis, and L. Reichel -- Partial Eigenvalue assignment for large linear control systems
• H. Faßbender and P. Benner -- A hybrid method for the numerical solution of discrete-time algebraic Riccati equations

• A. Böttcher and S. Grudsky -- Condition numbers of large Toeplitz-like matrices
• D. Fasino and V. Olshevsky -- How bad are symmetric Pick matrices?
• M. Fiedler -- Spectral properties of real Hankel matrices
• L. Elsner and S. Friedland -- Conjectures and remarks on the limit of the spectral radius of nonnegative and block Toeplitz matrices

• G. Heinig and V. Olshevsky -- The Schur algorithm for matrices with Hessenberg displacement structure
• Y. Eidelman and I. Gohberg -- Fast inversion algorithms for a class of block structured matrices
• S. Chandrasekaran and M. Gu -- A fast and stable solver for recursively semi-separable systems of linear equations

• M. Stewart -- Stability properties of several variants of the unitary Hessenberg $QR$ algorithm
• M. Kim, H. Park, and L. Eldén -- Comparison of algorithms for Toeplitz least squares and symmetric positive definite linear systems
• G. Heinig -- Stability of Toeplitz matrix inversion formulas
• J. Demmel and P. Koev -- Necessary and sufficient conditions for accurate and efficient rational function evaluation and factorizations of rational matrices
• M. Van Barel and A. Bultheel -- Updating and downdating of orthonormal polynomial vectors and some applications
• P. C. Hansen and P. Yalamov -- Rank-revealing decompositions of symmetric Toeplitz matrices

• R. H. Chan, M. K. Ng, and A. M. Yip -- A survey of preconditioners for ill-conditioned Toeplitz systems
• D. Potts and G. Steidl -- Preconditioning of Hermitian block-Toeplitz-Toeplitz-block matrices by level-1 preconditioners

• D. A. Bini and B. Meini -- Approximate displacement rank and applications
• W. F. Trench -- Properties of some generalizations of Kac-Murdock-Szegö matrices
• G. Heinig and K. Rost -- Efficient inversion formulas for Toeplitz-plus-Hankel matrices using trigonometric transformations
• L. Gemignani -- On a generalization of Poincaré's theorem for matrix difference equations arising from root-finding problems
• L. Rodman -- Completions of triangular matrices: A survey of results and open problems
• S. S. Capizzano and C. T. Possio -- Positive representation formulas for finite difference discretizations of (elliptic) second order PDEs
• P. Tilli -- On some problems involving invariant norms and Hadamard products
• Y. S. Choi, I. Koltracht, and P. J. McKenna -- A generalization of the Perron-Frobenius theorem for non-linear perturbations of Stiltjes matrices
• M. J. C. Gover and A. M. Byrne -- The Rhombus matrix: Definition and properties