Matrix Calculus (2nd Edition) by E. Bodewig
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Matrix Calculus (2nd Edition) written by
E. Bodewig .
The aim of this book is a systematic calculation with the true building blocks of a matrix, the rows and columns, avoiding the use of the individual elements.
Historically the notation of the theory of matrices developed from that of the theory of determinants, although the two subjects have little in common. This little was, however, enough to cause the complete notation of the highly developed theory of determinants, a notation which had proved very efficient and convenient for its own purposes, to be taken over by the new theory of matrices.
Thus the matrix, forced into the Procrustean bed of determinants, was broken down into its individual elements, although the elements themselves seldom play an independent part; indeed, the elements nearly always occur in fixed aggregates in which the individual elements themselves are without significance. The result of this operation with symbols foreign to the subject was often an awkward and unintelligible formula, containing, for example, several summation signs. From such a formula the method of calculation had to be derived more or less by a process of mentally grouping the various elements into rows and columns, a process which should already have been carried out by the notation itself. Thus there arose a discrepancy between thought and calculation, and lack of elegance was a sign of it.
Matrix Calculus (2nd Edition) written by
E. Bodewig
cover the following topics.
PART 1. MATRIX CALCULUS
1. Vectors
2. Matrices
3. Further Applications
4. treasures of the Magnitude of a Matrix
5. Forms
6. Eigenvalues
PART 11. LINEAR EQUATIONS
A.DIRECT METHODS
1. Exact Solutions
2. Approximate Solutions HO
B ITERATION METHODS
3. Dei elopment of Iteration Methods
4. Iteration I II
5. Characteristic Equation of Iteration Processes
6. Type of Convergence of Iteration Methods
7. Coniergence Theorems
8. General Iteration
9. Methods for Automatic Machines
10. Speeding up Convergence b\ Changing Matrix
11. The Iterated Direct Methods
12. Methods for Electronic Computers
13. yanous Questions
PART III. INVERSION OF MATRICES.
A.DIRECT METHODS
1. Condensation
2. Frobenius’s Relation
3. Completing
4. The Adjugate
b.ITERATION METHOD
C.GEODETIC matrices
Part IV.EIGENPROBLEMS
1. Introductoion*
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 Jlatrix
8. Deflation
9. Rutishauser s LR Algorithm
b.DIRECT METHODS
10. Determination of Eigen\ectors
11. Pure Methods
12. Progressne Algonthms
13. The Eigenproblem (A + jlB)x — 0
14. Special Matrices
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