Matrix Calculus - 3rd Edition - ISBN: 9781483232140, 9781483274980

Matrix Calculus

3rd Edition

Authors: E. Bodewig
eBook ISBN: 9781483274980
Imprint: North Holland
Published Date: 1st January 1959
Page Count: 466
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Description

Matrix Calculus, Second Revised and Enlarged Edition focuses on systematic calculation with the building blocks of a matrix and rows and columns, shunning the use of individual elements.

The publication first offers information on vectors, matrices, further applications, measures of the magnitude of a matrix, and forms. The text then examines eigenvalues and exact solutions, including the characteristic equation, eigenrows, extremum properties of the eigenvalues, bounds for the eigenvalues, elementary divisors, and bounds for the determinant.

The text ponders on approximate solutions, as well as the decomposition of the matrix into two triangular matrices, choice of another pivotal element, Gauss-Doolittle process, Aitken's triple product, neighbor systems, errors and exactness of the solution, and complex systems. The publication also elaborates on the characteristic equation of the iteration processes, type of convergence of the iteration methods, speeding-up convergence by changing matrix, and methods for electronic computers. The determination of eigenvectors, pure methods, progressive algorithms, and deflation are also discussed.

The manuscript is a helpful reference for researchers interested in matrix calculus.

Table of Contents


Part I. Matrix Calculus

1. Vectors

1.1. Equation of a Plane

2. Matrices

3. Further Applications

4. Measures of the Magnitude of a Matrix

5. Forms

6. Eigenvalues

6.1. Modal Matrix, Spectral Matrix

6.2. Characteristic Equation

6.3. Relations

6.4. Eigenrows

6.5. Extremum Properties

6.6. Bounds for Eigenvalues

6.7. Bounds for the Determinant

6.8. Elementary Divisors

Part II. Linear Equations

A. Direct Solutions

1. Exact Solutions

.2 Approximate Solutions

B. Iteration Methods

3.1. Introduction

3.2. Preliminary View

3.3. Development of Iteration Methods

4. Iteration I, II

5. Characteristic Equation of Iteration Processes

6. Type of Convergence of Iteration Methods

7. Convergence Theorems

8. General Iteration

9. Methods for Automatic Machines

10. Speeding-up Convergence by Changing Matrix

11. The Iterated Direct Methods

12. Methods for Electronic Computers

13. Various Questions

Part III. Inversion of Matrices

A. Direct Methods

1. Condensation

2. Frobenius's Relation

3. Completing

4. TheAdjugate

B. Iteration Method

C. Geodetic Matrices

Part IV. Eigenproblems

1. Introductory

A. Iteration Methods

2. The Iterated Vectors (Power Method)

3. Orthogonal Transformations

4. Method of Solving Linear Equations

5. The Gradient Method

6. The Use of Polynomials

7. Powers of the Matrix

8. Deflation

9. Rutishauser's LR-Algorithm

B. Direct Methods

10. Determination of Eigenvectors

11. Pure Methods

12. Progressive Algorithms

13. The Eigenproblem (A + λB)x = 0

14. Special Matrices

Exercises

Literature

Index

Details

No. of pages:
466
Language:
English
Copyright:
© North Holland 1959
Published:
Imprint:
North Holland
eBook ISBN:
9781483274980

About the Author

E. Bodewig