Ebook: A Polynomial Approach to Linear Algebra

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. Finally there is a chapter on Hankel norm approximation for the case of scalar rational functions which allows the reader to access ideas and results on the frontier of current research.

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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. Finally there is a chapter on Hankel norm approximation for the case of scalar rational functions which allows the reader to access ideas and results on the frontier of current research.
Content:
Front Matter....Pages i-xiii
Preliminaries....Pages 1-32
Linear Spaces....Pages 33-52
Determinants....Pages 53-64
Linear Transformations....Pages 65-96
The Shift Operator....Pages 97-126
Structure Theory of Linear Transformations....Pages 127-149
Inner Product Spaces....Pages 151-185
Quadratic Forms....Pages 187-249
Stability....Pages 251-269
Elements of System Theory....Pages 271-303
Hankel Norm Approximation....Pages 305-352
Back Matter....Pages 353-361

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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. Finally there is a chapter on Hankel norm approximation for the case of scalar rational functions which allows the reader to access ideas and results on the frontier of current research.
Content:
Front Matter....Pages i-xiii
Preliminaries....Pages 1-32
Linear Spaces....Pages 33-52
Determinants....Pages 53-64
Linear Transformations....Pages 65-96
The Shift Operator....Pages 97-126
Structure Theory of Linear Transformations....Pages 127-149
Inner Product Spaces....Pages 151-185
Quadratic Forms....Pages 187-249
Stability....Pages 251-269
Elements of System Theory....Pages 271-303
Hankel Norm Approximation....Pages 305-352
Back Matter....Pages 353-361
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