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Mathematics - Algebra | A Polynomial Approach to Linear Algebra

A Polynomial Approach to Linear Algebra

Series: Universitext

Fuhrmann, Paul A.

2nd ed. 2012, XVI, 411p.

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  • Many additions including Rational interpolation, interpolation using H^{\nfty} functions and tensor products and models (scalar case)
  • Almost 100 new pages added
  • Allows the reader to access ideas and results on the frontier of current research

A Polynomial Approach to Linear Algebra is a text which is heavily biased towards functional methods. In using the shift operator as a central object, it makes linear algebra a perfect introduction to other areas of mathematics, operator theory in particular. This technique is very powerful as becomes clear from the analysis of canonical forms (Frobenius, Jordan). It should be emphasized that these functional methods are not only of great theoretical interest, but lead to computational algorithms. Quadratic forms are treated from the same perspective, with emphasis on the important examples of Bezoutian and Hankel forms. These topics are of great importance in applied areas such as signal processing, numerical linear algebra, and control theory. Stability theory and system theoretic concepts, up to realization theory, are treated as an integral part of linear algebra.

This new edition has been updated throughout, in particular new sections  have been added on rational interpolation, interpolation using H^{\nfty} functions, and tensor products of models.

Review from first edition:

“…the approach pursued by the author is of unconventional beauty and the material covered by the book is unique.” (Mathematical Reviews, A. Böttcher)

Content Level » Professional/practitioner

Keywords » Elements of Model reduction - Linear Algebra - Rational interpolation - Tensor products and models

Related subjects » Algebra - Applications - Mathematics

Table of contents / Sample pages 

Preliminaries.- Linear Spaces.- Determinants.- Linear Transformations.- The Shift Operator.- Structure Theory of Linear Transformations.- Inner Product Spaces.- Quadratic Forms.- Stability.- Elements of System Theory.- Hankel Norm Approximation.

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