The theory of matrices,

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Preface......Page 4
PUBLISHERS' PREFACE......Page 7
Contents......Page 8
1. Matrices. Basic notation......Page 12
2. Addition and multiplication of rectangular matrices......Page 14
3. Square matrices......Page 23
4. Compound matrices. Minors of the inverse matrix......Page 30
1. Gauss's elimination method......Page 34
2. Mechanical interpretation of Gauss's algorithm......Page 39
3. Sylvester's determinant identity......Page 42
4. The decomposition of a square matrix into triangular factors......Page 44
5. The partition of a matrix into blocks. The technique of operating with partitioned matrices. The generalized algorithm of Gauss......Page 52
1. Vector spaces......Page 61
2. A linear operator mapping an ndimensional space into an mdimensional space......Page 66
3. Addition and multiplication of linear operators......Page 68
4. Transformation of coordinates......Page 70
5. Equivalent matrices. The rank of an operator. Sylvester's  inequality......Page 72
6. Linear operators mapping an ndimensional space into itself......Page 77
7. Characteristic values and characteristic vectors of a linear operator......Page 80
8. Linear operators of simple structure.......Page 83
1. Addition and multiplication of matrix polynomials......Page 87
2. Right and left division of matrix polynomials.......Page 88
3. The generalized Bezout theorem......Page 91
4. The characteristic polynomial of a matrix. The adjoint matrix......Page 93
5. The method of Faddeev for the simultaneous computation of the coefficients of the characteristic polynomial and of the ad joint matrix......Page 98
6. The minimal polynomial of a matrix.......Page 100
1. Definition of a function of a matrix......Page 106
2. The LagrangeSylvester interpolation polynomial......Page 112
3. Other forms of the definition of f (A). The components of the matrix A......Page 115
4. Representation of functions of matrices by means of series......Page 121
5. Application of a function of a matrix to the integration of a system of linear differential equations with constant coefficients......Page 127
6. Stability of motion in the case of a linear system......Page 136
1. Elementary transformations of a polynomial matrix.......Page 141
2. Canonical form of a Amatrix.......Page 145
3. Invariant polynomials and elementary divisors of a polynomial matrix......Page 150
4. Equivalence of linear binomials......Page 156
5. A criterion for similarity of matrices......Page 158
6. The normal forms of a matrix.......Page 160
7. The elementary divisors of the matrix f(A)......Page 164
8. A general method of constructing the transforming matrix......Page 170
9. Another method of constructing a transforming matrix......Page 175
1. The minimal polynomial of a vector and a space (with respect to a given linear operator)......Page 186
2. Decomposition into invariant subspaces with coprime minimal polynomials......Page 188
3. Congruence. Factor space......Page 192
4. Decomposition of a space into cyclic invariant subspaces.......Page 195
5. The normal form of a matrix......Page 201
6. Invariant polynomials. Elementary divisors......Page 204
7. The Jordan normal form of a matrix......Page 211
8. Krylov's method of transforming the secular equation......Page 213
1. The equation AX = XB......Page 226
2. The special case A = B. Commuting matrices.......Page 231
4. The scalar equation f (X) = 0......Page 236
5. Matrix polynomial equations......Page 238
6. The extraction of mtb roots of a nonsingular matrix.......Page 242
7. The extraction of mth roots of a singular matrix......Page 245
8. The logarithm of a matrix.......Page 250
1. General considerations......Page 253
2. Metrization of a space......Page 254
3. Gram's criterion for linear dependence of vectors......Page 257
4. Orthogonal projection......Page 259
5. The geometrical meaning of the Gramian and some inequalities......Page 261
6. Orthogonalization of a sequence of vectors......Page 267
7. Orthonormal bases......Page 273
8. The adjoint operator......Page 276
9. Normal operators in a unitary space......Page 279
10. The spectra of normal, hermitian, and unitary operators......Page 281
11. Positivesemidefinite and positivedefinite hermitian operators......Page 285
12. Polar decomposition of a linear operator in a unitary space. Cayley's formulas......Page 287
13. Linear operators in a euclidean space......Page 291
14. Polar decomposition of an operator and the Cayley formulas in a euclidean space......Page 297
15. Commuting normal operators......Page 301
1. Transformation of the variables in a quadratic form......Page 305
2. Reduction of a quadratic form to a sum of squares. The law of inertia......Page 307
3. The methods of Lagrange and Jacobi of reducing a quadratic form to a sum of squares......Page 310
4. Positive quadratic forms......Page 315
5. Reduction of a quadratic form to principal axes......Page 319
6. Pencils of quadratic......Page 321
7. Extremal properties of the characteristic values of a regular pencil of forms......Page 328
S. Small oscillations of a system with n degrees of freedom......Page 337
9. Hermitian forms......Page 342
10. Hankel forms......Page 349
BIBLIOGRAPHY......Page 362
INDEX......Page 380
Back Cover......Page 386