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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 becom...

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Bibliographic Details
Main Author: Fuhrmann, Paul A.
Format: eBook
Language:English
Published: New York, NY Springer New York 1996, 1996
Edition:1st ed. 1996
Series:Universitext
Subjects:
Calculus Of Variations And Optimization
Control Theory
Systems Theory, Control
Linear Algebra
System Theory
Algebras, Linear
Mathematical Optimization
Calculus Of Variations
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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
  • 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
  • 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

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