Matrix Calculus

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.

Inhalt
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 + B)x = 0


14. Special Matrices


Exercises


Literature


Index

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