Fuhrmann A Polynomial Approach to Linear Algebra

Erscheinungsjahr 2012
ISBN: 978-1-4419-8734-1
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 361 Seiten, eBook

Reihe: Universitext

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.

Zielgruppe

Graduate

Autoren/Hrsg.

Fuhrmann, Paul A.

Inhaltsverzeichnis

1 Preliminaries.- 1.1 Maps.- 1.2 Groups.- 1.3 Rings and Fields.- 1.4 Modules.- 1.5 Exercises.- 1.6 Notes and Remarks.- 2 Linear Spaces.- 2.1 Linear Spaces.- 2.2 Linear Combinations.- 2.3 Subspaces.- 2.4 Linear Dependence and Independence.- 2.5 Subspaces and Bases.- 2.6 Direct Sums.- 2.7 Quotient Spaces.- 2.8 Coordinates.- 2.9 Change of Basis Transformations.- 2.10 Lagrange Interpolation.- 2.11 Taylor Expansion.- 2.12 Exercises.- 2.13 Notes and Remarks.- 3 Determinants.- 3.1 Basic Properties.- 3.2 Cramer’s Rule.- 3.3 The Sylvester Resultant.- 3.4 Exercises.- 3.5 Notes and Remarks.- 4 Linear Transformations.- 4.1 Linear Transformations.- 4.2 Matrix Representations.- 4.3 Linear Punctionals and Duality.- 4.4 The Adjoint Transformation.- 4.5 Polynomial Module Structure on Vector Spaces.- 4.6 Exercises.- 4.7 Notes and Remarks.- 5 The Shift Operator.- 5.1 Basic Properties.- 5.2 Circulant Matrices.- 5.3 Rational Models.- 5.4 The Chinese Remainder Theorem.- 5.5 Hermite Interpolation.- 5.6 Duality.- 5.7 Reproducing Kernels.- 5.8 Exercises.- 5.9 Notes and Remarks.- 6 Structure Theory of Linear Transformations.- 6.1 Cyclic Transformations.- 6.2 The Invariant Factor Algorithm.- 6.3 Noncychc Transformations.- 6.4 Diagonalization.- 6.5 Exercises.- 6.6 Notes and Remarks.- 7 Inner Product Spaces.- 7.1 Geometry of Inner Product Spaces.- 7.2 Operators in Inner Product Spaces.- 7.3 Unitary Operators.- 7.4 Self-Adjoint Operators.- 7.5 Singular Vectors and Singular Values.- 7.6 Unitary Embeddings.- 7.7 Exercises.- 7.8 Notes and Remarks.- 8 Quadratic Forms.- 8.1 Preliminaries.- 8.2 Sylvester’s Law of Inertia.- 8.3 Hankel Operators and Forms.- 8.4 Bezoutians.- 8.5 Representation of Bezoutians.- 8.6 Diagonalization of Bezoutians.- 8.7 Bezout and Hankel Matrices.- 8.8 Inversion of HankelMatrices.- 8.9 Continued Fractions and Orthogonal Polynomials.- 8.10 The Cauchy Index.- 8.11 Exercises.- 8.12 Notes and Remarks.- 9 Stability.- 9.1 Root Location Using Quadratic Forms.- 9.2 Exercises.- 9.3 Notes and Remarks.- 10 Elements of System Theory.- 10.1 Introduction.- 10.2 Systems and Their Representations.- 10.3 Realization Theory.- 10.4 Stabilization.- 10.5 The Youla-Kucera Parametrization.- 10.6 Exercises.- 10.7 Notes and Remarks.- 11 Hankel Norm Approximation.- 11.1 Introduction.- 11.2 Preliminaries.- 11.3 Schmidt Pairs of Hankel Operators.- 11.4 Duality and Hankel Norm Approximation.- 11.5 Nevanhnna-Pick Interpolation.- 11.6 Hankel Approximant Singular Values.- 11.7 Exercises.- 11.8 Notes and Remarks.- Reference.