"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. It should be emphasized that these functional methods are not only of great theoretical interest, but lead to computational algorithms.
Reihe
Sprache
Verlagsort
Verlagsgruppe
Illustrationen
1
1 s/w Abbildung
XIII, 361 p. 1 illus.
Dateigröße
ISBN-13
978-1-4419-8734-1 (9781441987341)
DOI
10.1007/978-1-4419-8734-1
Schweitzer Klassifikation
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.
