Matrix Calculus (eBook)

Front Cover 1
Matrix Calculus 4
Copyright 
5 
Table of Contents 6
Dedication 3
PREFACE 10
PART I: MATRIX CALCULUS 14
CHAPTER I. VECTORS 16
1.1. EQUATION OF A PLANE 18
CHAPTER II. MATRICES 20
CHAPTER 3. FURTHER APPLICATIONS 44
CHAPTER 4. MEASURES OF THE MAGNITUDE OF A MATRIX 56
CHAPTER 5. FORMS 60
CHAPTER 6. EIGENVALUES 67
6.1. MODAL-MATRIX, SPECTRAL-MATRIX 67
6.2. THE CHARACTERISTIC EQUATION 71
6.3. RELATIONS BETWEEN Sp, N, |A|, 
74 
6.4. EIGENROWS 75
6.5. EXTREMUM PROPERTIES OF THE EIGENVALUES 76
6.6. BOUNDS FOR THE EIGENVALUES 80
6.7. BOUNDS FOR THE DETERMINANT 91
6.8. ELEMENTARY DIVISORS 95
PART II: LINEAR EQUATIONS 112
A. DIRECT METHODS 114
CHAPTER 1. EXACT SOLUTIONS 114
1.1. ELIMINATION I 115
1.2. ELIMINATION II 120
CHAPTER 2. APPROXIMATE SOLUTIONS 123
2.1. CONDENSATION I. TRIANGULARISATION 123
2.2. CONDENSATION II. DIAGONALIZATION 128
2 . 3 . THE DECOMPOSITION OF THE MATRIX INTO TWO TRIANGULAR MATRICES 129
2.4. CHOICE OF ANOTHER PIVOTAL ELEMENT 131
2.5. THE GAUSS-DOOLITTLE PROCESS 132
2.6. A METHOD FOR PUNCHED CARDS 141
2.7. THE GENERALIZED CONDENSATIONS I AND II 143
2.8. AlTKENS TRIPLE PRODUCT 145
2.9. ILL-CONDITIONED EQUATIONS 146
2.10. NEIGHBOUR SYSTEMS 150
2.11. ERRORS AND EXACTNESS OF THE SOLUTION 151
2.12. COMPLEX SYSTEMS 155
B. ITERATIONS METHODS 156
CHAPTER 3. 156
3.1. INTRODUCTION 156
3.2. PRELIMINARY VIEW 157
3.3. DEVELOPMENT OF THE ITERATION METHODS 157
CHAPTER 4. ITERATION I 163
CHAPTER 5. THE CHARACTERISTIC EQUATION OF THE ITERATION PROCESSES 170
CHAPTER 6. TYPE OF CONVERGENCE OF THE ITERATION METHODS 172
CHAPTER 7. CONVERGENCE THEOREMS 174
7.1. SCHMIDT-MISES-GEIRINGER 174
7.3. ITERATION II 176
7.4. ITERATION I 178
7.5. GEIRINGER'S THEOREM 178
7.6. THEOREM OF STEIN AND ROSENBERG 179
7.7. ANOTHER THEOREM OF STEIN-ROSENBERG 180
7.8. AITKEN'S NEO-SEIDELIAN ITERATION 181
CHAPTER 8. THE GENERAL ITERATION 184
CHAPTER 9. METHODS FOR AUTOMATIC MACHINES 188
CHAPTER 10. SPEEDING - U P CONVERGENCE BY CHANGING MATRIX 191
10.1. CESARl'S METHOD 191
10.2. VAN DER CORPUT'S DEVICE 192
10.3. THE METHOD OF ELIMINATION 193
10.4. JACOBl'S METHOD 194
CHAPTER 11. THE ITERATED DIRECT METHODS 196
11.1. CONVERGENCE OF THE METHOD 197
CHAPTER 12. METHODS FOR ELECTRONIC COMPUTERS 199
12.1. KACMARZ'S PROCEDURE 199
12.2. CIMMINO'S PROCEDURE 200
12.3. LINEAR EQUATIONS AS MINIMUM CONDITION 201
12.4. LINEAR EQUATIONS AS EIGENPROBLEMS 213
CHAPTER 13. VARIOUS QUESTIONS 215
13.1. NORMALIZATION 215
13.2. SCALING 215
13.3. ANOTHER SCALING 216
13.4. A THIRD SCALING 216
PART IIII: NVERSION OF MATRICES 218
A. DIRECT METHODS 220
CHAPTER 1. CONDENSATION 220
1.1. THE INVERSE OF A TRIANGULAR MATRIX 221
CHAPTER 2. FROBENIUS-SCHUR'S RELATION 230
CHAPTER 3. COMPLETING 236
CHAPTER 4. THE ADJUGATE 238
4 . 1 . THE METHOD OF DETERMINANTS 238
B. ITERATION METHOD 240
C. GEODETIC MATRICES 248
PART IV. EIGEN PROBLEMS 278
CHAPTER 1. INTRODUCTORY 280
A. ITERATION METHODS 282
CHAPTER 2. THE ITERATED VECTORS {Power Method) 282
2.1. THE DOMINANT EIGENVALUE IS REAL 283
2.2. THE DOMINANT EIGENVALUE IS COMPLEX 298
2.3. OTHER CASES 300
2.4. CRITICISM OF THE POWER METHOD 301
2.5. HIGHER EIGENVALUES 305
2.6. HIGHER EIGENVALUES ACCORDING TO AITKEN 332
2.7. THE LEAST EIGENVALUES 333
2.8. THE USE OF FROBENIUS'S THEOREM 334
2.9. WILKINSON'S METHOD 338
2.10. MULTIPLE EIGENVALUES 340
CHAPTER 3. ORTHOGONAL TRANSFORMATIONS 342
3.1. JACOBI'S METHOD 342
3.2. MAGNIER'S AND JAHN'S MODIFICATION 347
3.3. GIVENS'S METHOD 349
CHAPTER 4. THE METHOD OF SOLVING LINEAR EQUATIONS 354
4.1. THE ITERATION WITH A-1 355
CHAPTER 5. THE GRADIENT METHOD 358
CHAPTER 6. THE USE OF POLYNOMIALS 362
CHAPTER 7. POWERS OF THE MATRIX 366
CHAPTER 8. DEFLATION 370
8.1. HOTELLING'S DEFLATION 370
8.2. WIELANDT'S DEFLATION 373
8.3. A THIRD DEFLATION 377
8.4. A FOURTH DEFLATION 384
8.5. VECTOR-DEFLATION. SCALAR DEFLATION 385
8.6 WEDDERBURN'S THEOREM 389
8.7. MATRIX DEFLATION IN THE CASE OF HIGHER ELEMENTARY DIVISORS 391
CHAPTER 9. RUTISHAUSER'S LR-ALGORITHM 395
B. DIRECT METHODS 402
CHAPTER 10. DETERMINATION OF THE EIGENVECTORS 403
10.1. EIGENVALUES FOR NEIGHBOUR EQUATIONS 404
10.2. INFLUENCE OF MATRIX ERRORS 404
10.3. CHANGES OF THE ROOTS OF 
404 
10.4. THE NUMBER OF OPERATIONS 405
CHAPTER 11. PURE METHODS 408
11.1 LEVERRIER'S METHOD 408
11.2. KRYLOW-DUNCAN'S METHOD 410
11.3. SAMUELSON'S METHOD 410
11.4 OTHER METHODS 413
CHAPTER 12. PROGRESSIVE ALGORITHMS 414
12.1 A METHOD 414
12.2. LANCZOS'S 
414 
12.3. HESSENBERG'S METHOD 425
CHAPTER 13. THE EIGENPROBLEM (A + .B)x 
430 
CHAPTER 14. SPECIAL MATRICES 432
EXERCISES 453
LITERATURE 457
INDEX 462