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The mathematical modeling of problems of the real world often leads to problems in linear algebra involving structured matrices where the entries are defined by few parameters according to a compact formula. Matrix patterns and structural properties provide a uniform means for describing different features of the problem that they model. The analysis of theoretical and computational properties of these structures is a fundamental step in the design of efficient solution algorithms.
Certain structures are encountered very frequently and reflect specific features that are common to different problems arising in diverse fields of theoretical and applied mathematics and engineering. In particular, properties of shift invariance, shared by many mathematical entities like point-spread functions, integral kernels, probability distributions, convolutions, etc., are the common feature which originates Toeplitz matrices. In fact, Toeplitz matrices, characterized by having constant entries along their diagonals, are encountered in fields like image processing, signal processing, digital filtering, queueing theory, computer algebra, linear prediction and in the numerical solution of certain difference and differential equations, just to mention a few. The interest in this class of matrices is not motivated only by the applications; in fact, Toeplitz matrices are endowed with a very rich set of mathematical properties and there exists a very wide literature dated back to the first half of the last century on their analytic, algebraic, spectral and computational properties.
Other classes of structured matrices are less pervasive in terms of applications but nevertheless they are not less important. Frobenius matrices, Hankel matrices, Sylvester matrices and Bezoutians, encountered in control theory, in stability issues, and in polynomial computations have a rich variety of theoretical properties and have been object of many studies. Vandermonde matrices, Cauchy matrices, Loewner matrices and Pick matrices are more frequently encountered in the framework of interpolation problems.
Tridiagonal and more general banded matrices and their inverses, which are semiseparable matrices, are very familiar in numerical analysis. Their extension to more general classes and the design of efficient algorithms for them has recently received much attention.
Multi-dimensional problems lead to matrices which can be represented as structured block matrices with a structure within the blocks themselves.
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