Methods of Matrix Algebra

Name: Methods of Matrix Algebra
Brand: Academic Press
Price: 73.95 EUR
Availability: OnlineOnly

PEASE(Herausgeber*in)

Academic Press

Erschienen am 14. Mai 2014

425 Seiten

E-Book

PDF mit Wasserzeichen-DRM

Systemvoraussetzungen

978-0-08-095522-3 (ISBN)

73,95 €inkl. 7% MwSt.

Systemvoraussetzungen

für PDF mit Wasserzeichen-DRM

E-Book Einzellizenz

Als Download verfügbar

Beschreibung

Weitere Details

Weitere Ausgaben

Inhalt

Front Cover
Methods of Matrix Algebra
Copyright Page
Contents
Foreword
Symbols and Conventions
Chapter I. Vectors and Matrices
1. Vectors
2. Addition of Vectors and Scalar Multiplication
3. Linear Vector Spaces
4. Dimensionality and Bases
5. Linear Homogeneous Systems-Matrices
6. Partitioned Matrices
7. Addition of Matrices and Scalar Multiplication
8. Multiplication of a Matrix Times a Vector
9. Matrix Multiplication
10. An Algebra
11. Commutativity
12. Divisors of Zero
13. A Matrix as a Representation of an Abstract Operator
14. Other Product Relations
15. The Inverse of a Matrix
16. Rank of a Matrix
17. Gauss's Algorithm
18. 2-Port Networks
19. Example
Chapter II. The Inner Product
1. Unitary Inner Product
2. Alternative Representation of Unitary Inner Product
3. General (Proper) Inner Product
4. Euclidean Inner Product
5. Skew Axes
6. Orthogonality
7. Normalization
8. Gram-Schmidt Process
9. The Norm of a Vector
Chapter III. Eigenvalues and Eigenvectors
1. Basic Concept
2. Characteristic or Iterative Impedance
3. Formal Development
4. Determination of the Eigenvalues
5. Singularity
6. Linear Independence
7. Semisimplicity
8. Nonsemisimple Matrices
9. Degeneracy in a Chain
10. Examples
11. p-Section of a Filter
12. Structure of the Characteristic Equation
13. Rank of a Matrix
14. The Trace of a Matrix
15. Reciprocal Vectors
16. Reciprocal Eigenvectors
17. Reciprocal Generalized Eigenvectors
18. Variational Description of the Eigenvectors and Eigenvalues
Chapter IV. Hermitian, Unitary, and Normal Matrices
1. Adjoint Relation
2. Rule of Combination
3. The Basic Types
4. Decomposition into Hermitian Components
5. Polar Decomposition
6. Structure of Normal Matrices
7. The Converse Theorem
8. Hermitian Matrices
9. Unitary Matrices
10. General (Proper) Inner Product
Chapter V. Change of Basis, Diagonalization, and the Jordan Canonical Form
1. Change of Basis and Similarity Transformations
2. Equivalence Transformations
3. Congruent and Conjunctive Transformations
4. Example
5. Gauge Invariance
6. Invariance of the Eigenvalues under a Change of Basis
7. Invariance of the Trace
8. Variation of the Eigenvalues under a Conjunctive Transformation
9. Diagonalization
10. Diagonalization of Normal Matrices
11. Conjunctive Transformation of a Hermitian Matrix
12. Example
13. Positive Definite Hermitian Forms
14. Lagrange's Method
15. Canonical Form of a Nonsemisimple Matrix
16. Example
17. Powers and Polynomials of a Matrix
18. The Cayley-Hamilton Theorem
19. The Minimum Polynomiail
20. Examples
21. Summary
Chapter VI. Functions of a Matrix
1. Differential Equations
2. Reduction of Degree
3. Series Expansion
4. Transmission Line
5. Square Root Function
6. Unitary Matrices as Exponentials
7. Eigenvectors
8. Spectrum of a Matrix
9. Example
10. Commutativity
11. Functional Relations
12. Derivative of e-jRz, R Constant
13. R Not Constant
14. Derivative of a Power
15. Example
Chapter VII. The Matricant
1. Integral Matrix
2. The Matricant
3. Matricants of Related Systems
4. Peano Expansion
5. Error of Approximation
6. Bound on the Difference
7. Approximate Expansion
8. Inhomogeneous Equations
9. The Multiplicative Integral of Volterra
Chapter VIII. Decomposition Theorems and the Jordan Canonical Form
1. Decomposition
2. Decomposition into Eigensubspaces
3. Congruence and Factor Space
4. Cyclic Subspaces
5. Decomposition into Cyclic Subspaces
6. The Jordan Canonical Form
7. Invariant Polynomials and Elementary Divisors
8. Conclusions
Chapter IX. The Improper Inner Product
1. The Improper Inner Product
2. Failure of Cauchy-Schwartz and Triangle Inequalities
3. Orthogonal Sets of Vectors
4. Pairwise Orthogonality
5. Adjoint Operator
6. Orthogonality and Normality
7. K-Hermitian Matrices
8. K-Unitary Matrices
9. Orthogonalization of a Set of Vectors
10. Range of K
11. Determination of a Metric
Chapter X. The Dyad Expansion and Its Application
1. The Outer Product of Two Vectors
2. K-Dyads
3. Idempotency and Nilpotency
4. Expansion of an Arbitrary Matrix
5. Functions of a Matrix
6. Example
7. Differential Equations with Constant Coefficients
8. Perturbation Theory, Nondegenerate Case
9. Degenerate Case
10. Approximate Matricant
11. Conclusion
Chapter XI. Projectors
1. Definition of a Projector
2. Idempotency
3. Combinations of Projectors
4. Invariant Subspaces
5. Eigensubspaces
6. Semisimple Matrices
7. Nonsemisimple Matrices
8. Determination of P i , A Semisimple
9. A Not Semisimple
10. The Resolvant
11. Orthogonality
12. Conclusions
Chapter XII. Singular and Rectangular Operators
1. Abstract Formulation
2. Semisimple T
3. Example
4. Not Semisimple
5. Example
6. Conclusions
Chapter XIII. The Commutator Operator
1. Lie Groups
2. Infinitesimal Transformations of a Lie Group
3. Role of the Commutator
4. Lie Algebras
5. The Product Relation
6. K-Skew-Hermitian Algebra
7. The Ad Operator
8. Linearity
9. Eigenvalues and Eigenvectors
10. The Equation U = AdAV
11. The Killing Form
12. The Exponential of Ads
13. Simple Nonuniformity
14. The Exponentially Tapered Transmission Line
15. Conclusions
Chapter XIV. The Direct Product and the Kronecker Sum
1. The Direct Product
2. Justification of "Product"
3. The Product of Matricants and the Kronecker Sum
4. Group Theoretic Significance
5. Exponentiation
6. Eigenvectors and Eigenvalues of the Direct Product
7. Eigenvectors and Eigenvalues of the Kronecker Sum
8. Necessary Condition
9. Mixed Inverses
10. Summary
Chapter XV. Periodic Systems
1. Reducibility in the Sense of Lyapunov
2. Periodic Systems
3. Form of the Floquet Factors
4. Determination from the Matricant
5. The Floquet Modes
6. Space Harmonics
7. Orthogonality Relations
8. Example
9. Conclusions
Chapter XVI. Application to Electromagnetic Theory
1. Cartesian System
2. Maxwell's Equations
3. Magnetic Hertzian Vector Potential
4. Electric Hertzian Vector Potential
5. Change of Basis
6. Polarization Coordinates
7. Conrlusions
Chapter XVII. Sturm-Liouville Systems
1. Approximation in a Finitely Dimensioned Vector Space
2. Modified Sturm-Liouville Equation
3. The Characteristic Equation of H
4. Sturm Chains
5. Recursion Formula
Chapter XVIII. Markoff Matrices and Probability Theory
1. State Vector
2. Transition or Markoff Matrix
3. Eigenvectors of a Markoff Matrix
4. Reciprocal Eigenvectors
5. Nonnegative and Positive Matrices
6. Conclusions
Chapter XIX. Stability
1. The Basic Theorem for Stability
2. Routh-Hurwitz Method
3. Hurwitz Determinants
4. Criterion of Lienard and Chipart
5. Lyapunov's Second Method
6. A Metric as Lyapunov Matrix
7. Conclusions
References and Recommended Texts
Subject Index

Systemvoraussetzungen

Als PDF speichern Als Link merken