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Invariant Subspaces of Matrices with Applications 2nd Revised edition [Pehme köide]

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  • Formaat: Paperback / softback, 713 pages, kõrgus x laius x paksus: 229x152x27 mm, kaal: 780 g, illustrations
  • Sari: Classics in Applied Mathematics No. 51
  • Ilmumisaeg: 30-Jul-2006
  • Kirjastus: Society for Industrial & Applied Mathematics,U.S.
  • ISBN-10: 089871608X
  • ISBN-13: 9780898716085
Teised raamatud teemal:
Invariant Subspaces of Matrices with Applications 2nd Revised editionSuurem pilt
  • Formaat: Paperback / softback, 713 pages, kõrgus x laius x paksus: 229x152x27 mm, kaal: 780 g, illustrations
  • Sari: Classics in Applied Mathematics No. 51
  • Ilmumisaeg: 30-Jul-2006
  • Kirjastus: Society for Industrial & Applied Mathematics,U.S.
  • ISBN-10: 089871608X
  • ISBN-13: 9780898716085
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780898716085.html
Märksõnad:
This unique book addresses advanced linear algebra from a perspective in which invariant subspaces are the central notion and main tool. It contains comprehensive coverage of geometrical, algebraic, topological, and analytic properties of invariant subspaces. The text lays clear mathematical foundations for linear systems theory and contains a thorough treatment of analytic perturbation theory for matrix functions.Audience This book is appropriate for students, instructors, and researchers in applied linear algebra, linear systems theory, and signal processing. Its contents are accessible to readers who have had undergraduate-level courses in linear algebra and complex function theory.Contents Preface to the Classics Edition; Preface to the First Edition; Introduction; Part One: Fundamental Properties of Invariant Subspaces and Applications. Chapter 1: Invariant Subspaces: Definitions, Examples, and First Properties; Chapter 2: Jordan Form and Invariant Subspaces; Chapter 3: Coinvariant and Semiinvariant Subspaces; Chapter 4 Jordan Form for Extensions and Completions; Chapter 5: Applications to Matrix Polynomials; Chapter 6: Invariant Subspaces for Transformations Between Different Spaces; Chapter 7: Rational Matrix Functions; Chapter 8: Linear Systems; Part Two: Algebraic Properties of Invariant Subspaces. Chapter 9: Commuting Matrices and Hyperinvariant Subspaces; Chapter 10: Description of Invariant Subspaces and Linear Transformation with the Same Invariant Subspaces; Chapter 11: Algebras of Matrices and Invariant Subspaces; Chapter 12: Real Linear Transformations; Part Three: Topological Properties of Invariant Subspaces and Stability. Chapter 13: The Metric Space of Subspaces; Chapter 14: The Metric Space of Invariant Subspaces; Chapter 15: Continuity and Stability of Invariant Subspaces; Chapter 16: Perturbations of Lattices of Invariant Subspaces with Restrictions on the Jordan Structure; Chapter 17: Applications; Part Four: Analytic Properties of Invariant Subspaces. Chapter 18: Analytic Families of Subspaces; Chapter 19: Jordan Form of Analytic Matrix Functions; Chapter 20: Applications; Appendix: List of Notations and Conventions; References; Author Index; Subject Index.



This unique book addresses advanced linear algebra from a perspective in which invariant subspaces are the central notion and main tool. It contains comprehensive coverage of geometrical, algebraic, topological, and analytic properties of invariant subspaces. The text lays clear mathematical foundations for linear systems theory and contains a thorough treatment of analytic perturbation theory for matrix functions.

This unique book addresses advanced linear algebra using invariant subspaces as the central notion and main tool.

Muu info

This unique book addresses advanced linear algebra using invariant subspaces as the central notion and main tool.
Introduction 1(2)
Part One Fundamental Properties of Invariant Subspaces and Applications 3(290)
Chapter One Invariant Subspaces: Definition, Examples, and First Properties
5(40)
1.1 Definition and Examples
5(5)
1.2 Eigenvalues and Eigenvectors
10(2)
1.3 Jordan Chains
12(4)
1.4 Invariant Subspaces and Basic Operations on Linear Transformations
16(4)
1.5 Invariant Subspaces and Projectors
20(5)
1.6 Angular Transformations and Matrix Quadratic Equations
25(3)
1.7 Transformations in Factor Spaces
28(3)
1.8 The Lattice of Invariant Subspaces
31(6)
1.9 Triangular Matrices and Complete Chains of Invariant Subspaces
37(3)
1.10 Exercises
40(5)
Chapter Two Jordan Form and Invariant Subspaces
45(60)
2.1 Root Subspaces
45(7)
2.2 The Jordan Form and Partial Multiplicities
52(6)
2.3 Proof of the Jordan Form
58(2)
2.4 Spectral Subspaces
60(5)
2.5 Irreducible Invariant Subspaces and Unicellular Transformations
65(4)
2.6 Generators of Invariant Subspaces
69(3)
2.7 Maximal Invariant Subspace in a Given Subspace
72(6)
2.8 Minimal Invariant Subspace over a Given Subspace
78(5)
2.9 Marked Invariant Subspaces
83(2)
2.10 Functions of Transformations
85(7)
2.11 Partial Multiplicities and Invariant Subspaces of Functions of Transformations
92(3)
2.12 Exercises
95(10)
Chapter Three Coinvariant and Semiinvariant Subspaces
105(16)
3.1 Coinvariant Subspaces
105(4)
3.2 Reducing Subspaces
109(3)
3.3 Semiinvariant Subspaces
112(4)
3.4 Special Classes of Transformations
116(3)
3.5 Exercises
119(2)
Chapter Four Jordan Form for Extensions and Completions
121(23)
4.1 Extensions from an Invariant Subspace
121(7)
4.2 Completions from a Pair of Invariant and Coinvariant Subspaces
128(5)
4.3 The Sigal Inequalities
133(3)
4.4 Special Case of Completions
136(6)
4.5 Exercises
142(2)
Chapter Five Applications to Matrix Polynomials
144(45)
5.1 Linearizations, Standard Triples, and Representations of Monic Matrix Polynomials
144(9)
5.2 Multiplication of Monic Matrix Polynomials and Partial Multiplicities of a Product
153(3)
5.3 Divisibility of Monic Matrix Polynomials
156(5)
5.4 Proof of Theorem 5.3.2
161(6)
5.5 Example
167(4)
5.6 Factorization into Several Factors and Chains of Invariant Subspaces
171(4)
5.7 Differential Equations
175(5)
5.8 Difference Equations
180(3)
5.9 Exercises
183(6)
Chapter Six Invariant Subspaces for Transformations Between Different Spaces
189(23)
6.1 [ A B]-Invariant Subspaces
189(3)
6.2 Block Similarity
192(5)
6.3 Analysis of the Brunovsky Canonical Form
197(3)
6.4 Description of [ A B]-Invariant Subspaces
200(3)
6.5 The Spectral Assignment Problem
203(4)
6.6 Some Dual Concepts
207(2)
6.7 Exercises
209(3)
Chapter Seven Rational Matrix Functions
212(50)
7.1 Realizations of Rational Matrix Functions
212(6)
7.2 Partial Multiplicities and Multiplication
218(7)
7.3 Minimal Factorization of Rational Matrix Functions
225(5)
7.4 Example
230(4)
7.5 Minimal Factorizations into Several Factors and Chains of Invariant Subspaces
234(4)
7.6 Linear Fractional Transformations
238(6)
7.7 Linear Fractional Decompositions and Invariant Subspaces of Nonsquare Matrices
244(7)
7.8 Linear Fractional Decompositions: Further Deductions
251(4)
7.9 Exercises
255(7)
Chapter Eight Linear Systems
262(28)
8.1 Reductions, Dilations, and Transfer Functions
262(3)
8.2 Minimal Linear Systems: Controllability and Observability
265(5)
8.3 Cascade Connections of Linear Systems
270(4)
8.4 The Disturbance Decoupling Problem
274(5)
8.5 The Output Stabilization Problem
279(6)
8.6 Exercises
285(5)
Notes to Part 1.
290(3)
Part Two Algebraic Properties of Invariant Subspaces 293(92)
Chapter Nine Commuting Matrices and Hyperinvariant Subspaces
295(21)
9.1 Commuting Matrices
295(6)
9.2 Common Invariant Subspaces for Commuting Matrices
301(2)
9.3 Common Invariant Subspaces for Matrices with Rank 1 Commutators
303(2)
9.4 Hyperinvariant Subspaces
305(2)
9.5 Proof of Theorem 9.4.2
307(4)
9.6 Further Properties of Hyperinvariant Subspaces
311(2)
9.7 Exercises
313(3)
Chapter Ten Description of Invariant Subspaces and Linear Transformations with the Same Invariant Subspaces
316(23)
10.1 Description of Irreducible Subspaces
316(7)
10.2 Transformations Having the Same Set of Invariant Subspaces
323(5)
10.3 Proof of Theorem 10.2.1
328(10)
10.4 Exercises
338(1)
Chapter Eleven Algebras of Matrices and Invariant Subspaces
339(20)
11.1 Finite-Dimensional Algebras
339(1)
11.2 Chains of Invariant Subspaces
340(3)
11.3 Proof of Theorem 11.2.1
343(3)
11.4 Reflexive Lattices
346(4)
11.5 Reductive and Self-Adjoint Algebras
350(5)
11.6 Exercises
355(4)
Chapter Twelve Real Linear Transformations
359(25)
12.1 Definition, Examples, and First Properties of Invariant Subspaces
359(4)
12.2 Root Subspaces and the Real Jordan Form
363(3)
12.3 Complexification and Proof of the Real Jordan Form
366(5)
12.4 Commuting Matrices
371(3)
12.5 Hyperinvariant Subspaces
374(4)
12.6 Real Transformations with the Same Invariant Subspaces
378(2)
12.7 Exercises
380(4)
Notes to Part 2.
384(1)
Part Three Topological Properties of Invariant Subspaces and Stability 385(178)
Chapter Thirteen The Metric Space of Subspaces
387(36)
13.1 The Gap Between Subspaces
387(5)
13.2 The Minimal Angle and the Spherical Gap
392(4)
13.3 Minimal Opening and Angular Linear Transformations
396(4)
13.4 The Metric Space of Subspaces
400(6)
13.5 Kernels and Images of Linear Transformations
406(2)
13.6 Continuous Families of Subspaces
408(3)
13.7 Applications to Generalized Inverses
411(4)
13.8 Subspaces of Normed Spaces
415(5)
13.9 Exercises
420(3)
Chapter Fourteen The Metric Space of Invariant Subspaces
423(21)
14.1 Connected Components: The Case of One Eigenvalue
423(3)
14.2 Connected Components: The General Case
426(2)
14.3 Isolated Invariant Subspaces
428(4)
14.4 Reducing Invariant Subspaces
432(5)
14.5 Coinvariant and Semiinvariant Subspaces
437(2)
14.6 The Real Case
439(4)
14.7 Exercises
443(1)
Chapter Fifteen Continuity and Stability of Invariant Subspaces
444(38)
15.1 Sequences of Invariant Subspaces
444(3)
15.2 Stable Invariant Subspaces: The Main Result
447(4)
15.3 Proof of Theorem 15.2.1 in the General Case
451(4)
15.4 Perturbed Stable Invariant Subspaces
455(4)
15.5 Lipschitz Stable Invariant Subspaces
459(4)
15.6 Stability of Lattices of Invariant Subspaces
463(1)
15.7 Stability in Metric of the Lattice of Invariant Subspaces
464(4)
15.8 Stability of [ A B]-Invariant Subspaces
468(2)
15.9 Stable Invariant Subspaces for Real Transformations
470(5)
15.10 Partial Multiplicities of Close Linear Transformations
475(4)
15.11 Exercises
479(3)
Chapter Sixteen Perturbations of Lattices of Invariant Subspaces with Restrictions on the Jordan Structure
482(32)
16.1 Preservation of Jordan Structure and Isomorphism of Lattices
482(4)
16.2 Properties of Linear Isomorphisms of Lattices: The Case of Similar Transformations
486(6)
16.3 Distance Between Invariant Subspaces for Transformations with the Same Jordan Structure
492(5)
16.4 Transformations with the Same Derogatory Jordan Structure
497(3)
16.5 Proofs of Theorems 16.4.1 and 16.4.4
500(7)
16.6 Distance between Invariant Subspaces for Transformations with Different Jordan Structures
507(3)
16.7 Conjectures
510(3)
16.8 Exercises
513(1)
Chapter Seventeen Applications
514(47)
17.1 Stable Factorizations of Matrix Polynomials: Preliminaries
514(6)
17.2 Stable Factorizations of Matrix Polynomials: Main Results
520(5)
17.3 Lipschitz Stable Factorizations of Monic Matrix Polynomials
525(3)
17.4 Stable Minimal Factorizations of Rational Matrix Functions: The Main Result
528(4)
17.5 Proof of the Auxiliary Lemmas
532(5)
17.6 Stable Minimal Factorizations of Rational Matrix Functions: Further Deductions
537(3)
17.7 Stability of Linear Fractional Decompositions of Rational Matrix Functions
540(5)
17.8 Isolated Solutions of Matrix Quadratic Equations
545(6)
17.9 Stability of Solutions of Matrix Quadratic Equations
551(2)
17.10 The Real Case
553(4)
17.11 Exercises
557(4)
Notes to Part 3.
561(2)
Part Four Analytic Properties of Invariant Subspaces 563(83)
Chapter Eighteen Analytic Families of Subspaces
565(39)
18.1 Definition and Examples
565(4)
18.2 Kernel and Image of Analytic Families of Transformations
569(6)
18.3 Global Properties of Analytic Families of Subspaces
575(3)
18.4 Proof of Theorem 18.3.1 (Compact Sets)
578(6)
18.5 Proof of Theorem 18.3.1 (General Case)
584(6)
18.6 Direct Complements for Analytic Families of Subspaces
590(4)
18.7 Analytic Families of Invariant Subspaces
594(2)
18.8 Analytic Dependence of the Set of Invariant Subspaces and Fixed Jordan Structure
596(3)
18.9 Analytic Dependence on a Real Variable
599(2)
18.10 Exercises
601(3)
Chapter Nineteen Jordan Form of Analytic Matrix Functions
604(20)
19.1 Local Behaviour of Eigenvalues and Eigenvectors
604(3)
19.2 Global Behaviour of Eigenvalues and Eigenvectors
607(6)
19.3 Proof of Theorem 19.2.3
613(3)
19.4 Analytic Extendability of Invariant Subspaces
616(4)
19.5 Analytic Matrix Functions of a Real Variable
620(2)
19.6 Exercises
622(2)
Chapter Twenty Applications
624(21)
20.1 Factorization of Monic Matrix Polynomials
624(3)
20.2 Rational Matrix Functions Depending Analytically on a Parameter
627(7)
20.3 Minimal Factorizations of Rational Matrix Functions
634(5)
20.4 Matrix Quadratic Equations
639(3)
20.5 Exercises
642(3)
Notes to Part 4.
645(1)
Appendix. Equivalence of Matrix Polynomials 646(33)
A.1 The Smith Form: Existence
646(5)
A.2 The Smith Form: Uniqueness
651(3)
A.3 Invariant Polynomials, Elementary Divisors, and Partial Multiplicities
654(5)
A.4 Equivalence of Linear Matrix Polynomials
659(3)
A.5 Strict Equivalence of Linear Matrix Polynomials: Regular Case
662(4)
A.6 The Reduction Theorem for Singular Polynomials
666(6)
A.7 Minimal Indices and Strict Equivalence of Linear Matrix Polynomials (General Case)
672(6)
A.8 Notes to the Appendix
678(1)
List of Notations and Conventions 679(4)
References 683(4)
Author Index 687(2)
Subject Index 689


Israel Gohberg is Professor Emeritus at Tel-Aviv University and Free University of Amsterdam, and Dr. Honoris Causa at several European universities. He has contributed to the fields of functional analysis and operator theory, systems theory, matrix analysis and linear algebra, and computational techniques for integral equations and structured matrices. He has coauthored 25 books. Peter Lancaster is a Faculty Professor and Professor Emeritus at the University of Calgary and Honorary Research Fellow of the University of Manchester. He has published prolifically in matrix and operator theory and in many of their applications in numerical analysis, mechanics, and other fields. Leiba Rodman is Professor of Mathematics at the College of William and Mary and has done extensive work in matrix analysis, operator theory, and related fields. He has authored one book and coauthored six others.