Matrix calculator online

Matrix Theory
Home edition of Physical-Mathematical Literature Publishing House "Nauka", Moscow, 1973, 280 pp.
The book is intended to be the basis for courses and handbook
for all those interested in applied aspects of matrix theory. It can
seen as a good addition to the normal course of linear algebra
(the first two chapters - a statement of linear algebra in the matrix language).
A rigorous exposition of the theory of matrices it is combined with discussion
applied issues, partly classical, partly new.
CONTENTS
Preface 7
Chapter 1
Linear spaces, matrix algebra and linear algebraic
equation
1.1. Bound vectors in three-dimensional space 9
1.2. Space Rn and Cn 11
1.3. Interior Artworks 14
1.4. Linear combinations of 16
1.5. Matrix algebra 17
1.6 Partitions of matrices 20
1.7. Column vectors and vector-line 23
1.8. Annihilated subspace and the range of 25
1.9 Linear dependence and dimension 27
1.10. Properties of the basis vectors 32
1.11. Function definition of the determinant 34
1.12. Properties of determinants 36
1.13. Adjoint and the inverse matrix of 39
1.14. Binet formula - Cauchy 41
1.15. The rank of matrix 44
1.16. Solution of equations 48
1.17. Cramer's rule 51
Mixed exercises 52
Chapter 2
Eigenvalues and eigenvectors
2.1. Characteristic equation 54
2.2. Multiplicity eigenvalue 57
2.3. Eigenvectors 58
2.4. Similarity transformations and simple matrix 59
2.5. Spectral theorem, and polynomials of matrices 63
2.6. Orthogonal vectors and Quasiorthogonal 67
2.7. Orthonormal system 69
2.8. Special types of matrices 73
2.9. Hermitian matrix 75
2.10. Unitary such transformations 79
2.11. Idempotent Boolean matrices and projection 82
2.12. Hermitian and quadratic forms 85
2.13. Method of bringing Lagrange 90
2.14. Certain matrix 93
2.15. Theory of small oscillations and simultaneous application of quadratic 98
Forms
2.16. Fluctuations with external forces 102
Mixed exercises 103
Chapter 3
Variational method
3.1. Introduction 106
3.2. Extremal eigenvalues and the Rayleigh ratio of 106
3.3. The property of stationarity of the Rayleigh relations 108
3.4. Variational description of the eigenvalues 110
3.5. Problems with constraints 111
3.6. Theorem Courant - Fisher 113
3.7. Applications to the theory of small oscillations 118
Mixed exercises 120
Chapter 4
Minimal polynomial and normal forms
4.1. Introduction 121
4.2. Algebra λ-matrices with 122
4.3. λ-matrix with matrix arguments online calculator 125
4.4. Annihilating polynomials 128
4.5. The reduced adjacency matrices and the minimal polynomial 132
4.6. Elementary operations and equivalence of λ-matrices 134
4.7. Bringing λ-matrices equivalent transformations to 137
simplest form
4.8. Equivalent transformation matrices from Fnxn 140
4.9. Invariant polynomials and the canonical form of Smith 140
4.10. Similarity 143
4.11. The first natural normal form 144
4.12. Elementary divisors over a field of complex numbers 146
4.13. Second natural normal form and Jordan canonical 149
shape
Mixed exercises 153
Supplement to Chapter 4 154
Chapter 5
Functions of matrices
5.1. Introduction 156
5 2 Interpolation Polynomials 156
5 3. Defining the functions of the matrix 158
5.4. Spectral decomposition for f (A) 163
5.5. The properties of the component matrices 166
5.6. Sequences and series of matrices 170
5.7. Properties of some elementary functions 174
5.8. Using contour integrals 175
5.9. Applications to the solution of differential equations 178
Mixed exercises 182
Chapter 6
Norms of vectors and matrices
6.1. Matrix calculator online norm 185
6.2. Vector norms 190
6.3. Induced matrix norms 194
6.4. Absolute vector norms 199
6.5. Lower face 201
6.6. Field values 203
Chapter 7
Perturbation theory and estimates for the eigenvalues
7.1 Perturbations in solving linear equations 205
7.2. Theorem Gershgorin 203
7.3 Schur's theorem 212
7.4 Perturbation of eigenvalues of a simple matrix of 214
7.5. Analytic perturbation 219
7.6. Perturbation component matrices 220
7.7. Perturbation of repetition-free eigenvalue 223
7.8. Estimation of coefficients of the perturbation 224
7.9. The perturbation of multiple eigenvalue 227
7.10. Reducing process 232
Chapter 8
Direct products, solution of matrix equations and problems
stability
8.1. Introduction 231
8.2. The direct product of 235
8.3. Eigenvalues of composite matrices 237
8.4. Solution of linear matrix equations 239
8.5. Equation AX + XB = C 240
8.6. Fabric 242
8.7. Lyapunov Stability Theory 245
8.8. Criterion Routh - Hurwitz 248
Chapter 9
Nonnegative matrices
9.1. Introduction 255
9.2. Theorem of Perron - Frobenius 257
9.3. Reducible Matrix calculator online 262
9.4. Primitive and imprimitive matrix 264
9.5. Stochastic matrices 266
9.6. Markov Chains 268
Supplement 1. Some theorems from analysis of 271
Appendix 2. Generalized inverse matrix 273
Supplement 3. Recommendations for further reading 277
Index 278
INDEX
- Strongly connected 256
Algorithm of division 123
Greville (T. N. E. Grevill) 278
Associativity 10, 18
Grimshaw (ME Grimshaw) 277
Basis 27
Divisor left 123
Buck (R. S. Buck) 275
- Total 129
Bellman (R. Bellman) 279
- - The largest (GBR) 129
Bendixson (I. Bendixon) 213
- Rule 123
Block 21
- Elementary 147
Varga (R. S. Varga) 279
- - Line 148
The vector unit 11
- - Nonlinear calculator 148
- Normal form 101
Delta kroyekerovskaya 20
- Normalized 68
Diagonal Home 20
- Zero 9
Distributivity 18
- The order of M 14
Length 10. 14, 15
- Associated 9
Algebraic Supplement 36
- Own L 58
Extras 82
- - Limited 112
- Orthogonal 82, 112
- - R 54
The dependence of the linear 27
Vector-column 23
Target general Hermite interpolation
Vector-line 23
157
Vectors Biorthogonal 67
Sylvester's law of inertia 90
- Kvazibiortogonalnye 63, 68
Meaning own 54
- Quasiorthogonal 68
- - Limited 112
- Kvaziortonormirovannye 68
- Function on the spectrum of the matrix 159
- Orthogonal 14, 67
- Λ-matrix of left 125
- Orthonormal 67
Conditional probability 269 - - Rule 125
Type fluctuations 100 index of imprimitivity 264
Perturbation of a linear 227 - eigenvalue 154
- Relative 203 - form 90
Hamburger (N. L. Hamburger) 277 The integral of the vector 102
Hamilton (WR Hamilton) 126 - - matrix 176
Gantmacher FR 279 - Private 181
Harmonica normal 101 Kato (T. Kato) 228, 232, 277
Edge of the lower matrix 202 Quadrics Central 92
Graf sent and 256 Keli (A. Cayley) 126
- Special 40
Box 150 Jordan
- Commutator 87
The combination of a linear 16
- Reducible 255
Commutativity 10, 18
- Primitive 265
Component of the matrix 164
- Accession 39
Coordinates normal 101
- - Contained 129
Root hidden calculator online 147
- A simple 61
Cosine guide 10
- Rectangular 17
Cauchy (A. L. Cauchy) 116
- With a dominant diagonal 211
Geometric Multiplicity 58
- Symmetric 74
- Hidden root 149
- Scalar 19
- Eigenvalue 57
- Accompanying 144, 252
Criterion Gram 72
- Accompanying 64
- Routh-Hurwitz 249
- Stochastic 266
Courant (R. Courant) 111
- Transpose 17
Liapunov, AM, 245-247, 249
- Triangular 39
MacDuffie (SS MacDuffee) 241, 277
- Unitary 74
Marcus (M. Marcus) 277
- Stable 234, 245
Matrix 17
- Elementary 136
- Vandermonde 39
- Hermite 74, 228 Dies
- Home 41
Hurwitz 249
- Defective 61
- Commuting 18
- 20 diagonal
- Like 59
- - Canonical 137
- - 78 orthogonally
- Unit 19
- - 78 unitarily
- Idempotent 65
- Equivalent to 18
- Imprimitive 264
Mink (N. Mine) 277
Matrix square 17
Miior 38, 41
- Quasidiagonal 146
- Chief 41
- Skew-symmetric 74
Mirsky (L. Mirsky) 277
- Skew-Symmetric 74
Polynomial annihilating 128
- Polynomials 122
- Invariant 141
- Smooth 40
- Lagrange 157
- Nonnegative 255
- Minimum 128
- Irreducible 256
- Irreducible 129
- Nilpotent 66
- Contained 128
- Normal 80
- With a matrix argument 64
- Zero 19
- Characteristic 54
- Contact 40
Coprime polynomials 129
- - Generalized 273
Convex set 193
- Some 93
- Closed 271
- - Nonnegative 93
- Limited 271
- - Positive 93
Monotonicity absolute
- Orthogonal 74
Conversion 24, 60
vector norm 200
- Congruent 86
Independent linear 27
- Similarity 59
Schwarz inequality 15, 44
- Equivalent to 135, 136
Norma vector 190
Projection 15, 84
- - Absolute 199
The product of external 24
- Holder 190. 192
- Internal 14, 24
- Euclidean 189
- Direct 235
- Matrix 185
- Scalar 14
- - Induced 194
Derivative vector 99
- - Generalized 186
- Matrix 99, 176
- Spectral 186
- Determinant 52
- Frobenius 189
Infinite-dimensional spaces 28
The rules agreed 191
- Linear 11
The range of the matrix 25
- - Spanned by the vector system
Shell convex 193
16
Image matrix 25
- - Generated by the system
Neighborhood 271
vectors 16
Self-adjoint operator 76
- Own left 58
Operation binary 12
- - Rules 58
- - Closed 13
- Column 25
- Elementary left 135
Spectral radius 188
- - Rule 135
Partition 21
Osi quadric principal 93
- Square matrix
Ostrovsky (AM Ostroweki) 277
symmetrical 22
The ratio of Rayleigh 106
Expansion of the determinant on the column
- - Generalized 109
36
- Equivalence 60
- - - Line 36
Penrose (R. A. Penrose) 276
- Spectral 163
Shifter 34
Dimension 28
Perlis (S. Perils) 277
Reinhart (R. F. Rinehart) 277
Perron online(O. Perron) 262
The rank of matrix 46
Subspaces additional 82
- For minors of 46
Subspace 16
- - Column 45
- Cancellation of 25
- - Line 45
- Own 16
- Form 90
Algebraically closed field of 55
- Λ-matrix 140
- Values 203
The order of the matrix Routh 17 (E. J. Routh) 116
- 41 minor Resolvent 176
The sequence of matrices Rayleigh (Lord Raylegh) 116
diverging 170 Rellich (F. Rellich) 228
- - 170 converging Reflexivity similarity matrices 60
Rule 10 Decision parallelogram nontrivial 24
Wilkinson (J. H. Wilkinson) 277
- Approximate in the sense
Multiplication of matrices 17, 18
least quadratic
- The scalar 13, 17
deviation 275
Equation of nonuniform 24
Decision trivial 24
- Homogeneous 24
Rudin (W. Rudin) 273
- Characteristic 24
A number of matrices 172
Linear algebraic equations
- - Divergent 172
24
- - Convergent 172
Faddeev DK 277
- Puiseux 220
Faddeev and VN 277
Communication 112
Finkbeyier (D. T. Finkbeiner) 277
Segment line 193
Fischer (E. Fischer) 111
Signature 90
Smith canonical form 141
The symmetry of similarity of matrices 60
- 86 square online
Next 56
- Normal natural second
Addition of vectors 13
150
Smith (N. J. S. Smith) 141
- - - First 146
Spectrum of 63
- - Jordan 150
Stationarity relations Rayleigh
- Hermite 85
109
The degree of λ-matrix 122 Binet formula - Cauchy 41, 47
- Liouville 183
Sum of the vectors 10
- Orlando 249 (See http://online-calculator-for-free.blogspot.com/)
- Kronecker 238
Frobenius (G. Frobenius) 126, 141,
- Matrix 17
241, 261
- Line 82
Function 12
Sphere of unit 106, 192
- Additive 36
Theorem Apollo 16
- Analysis 175
- Gershgorin 209
- Continuous 271
- Cayley - Hamilton 126
- Homogeneous 36
- Courant - Fisher 115, 119
- Defined on the spectrum of 159
- The residue 176
- From the matrix 160
- - Parallelogram 16
- A whole 173
- The rest of 125
Halmos (P. R. Halmos) 277
- Perron - Frobenius 259-261
Householder (A. S. Householder) 277
- Pythagoras 15
Hirsch (K. A. Hirsch) 213
- Spectral 64
Hohn (F. E. Hohn) 277
- Shura 212
Markov chain 270
Teplitz (O. Toeplitz) 79
- - Homogeneous 270
Identity Cauchy 44
Circulant 244
- Jacobi 183
Private Left 123
Point limit 271
- Rule 123
Transitivity of similarity matrices 60
Frequencies of 100
Transposed 34
Number of conditional 207
Angle 10, 67
- - 207 spectral element matrix 17
Ball 193, 271 n-ca numbers ordered 9
- Unit 193 p-norm of 190
λ-matrix 122
Schneider (H. Schneider) 277
Schur (I. Schur) 79, 212 - regular 122
Equivalence of matrices 140
- Λ-matrix 136