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| Авторизация |
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| Поиск по указателям |
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| Bronson R. — Schaum's Outline of Matrix Operations |
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| Предметный указатель |
 (matrix exponential) 72 100
(matrix exponential) for a positive definite matrix 135
 , norm 110
 norm 110
 , norm 110
 norm 110
 , norm 110
Addition of matrices 2
Adjoint 119
Angle between vectors 143
Augmented matrix 11
AX + XB = C 73
AX = B see “Simultaneous linear equations”
AXB = C 166
Band matrix 160
Bessel function 80
Block matrix 43
Canonical basis 82
Canonical basis for a normal matrix 119
Cayley — Hamilton theorem 61
Chain 82
Characteristic equation 60
Characteristic equation of similar matrices 91
Characteristic polynomial 60
Characteristic value see “Eigenvalue”
Characteristic vector see “Eigenvector”
Cholesky decomposition 129
Circulant matrix 160
Coefficient matrix 11
Cofactor 42 50
Cofactor matrix 44
Column rank 53
Column vector 1
Companion matrix 70
Compatible norms 111 116
Complete pivoting 19
Complex conjugate 103
Complex conjugate transpose matrix 119
Complex matrix 1 103
Complex quadratic form 144 (see also “Quadratic form”)
Component 52
Condition number 118
Congruent matrices 144 151
Conjunctive matrices 145
Consistency condition for norms 110
Consistent simultaneous equations 11
Constant matrix 1
Convex combination 59
Crout’s reduction 25
Crout’s reduction for a tridiagonal matrix 161
Derivative of a matrix 72
Determinant 42
Determinant and eigenvalues 60
Determinant for a unitary matrix 136
Determinant of a positive definite matrix 132
Determinant of an LU decomposition 51
Diagonal 24 127
Diagonal matrix 24 127 137
Diagonal quadratic form 144
Differential equations 72
DIMENSION 52
Direct product 165
Distance between vectors 110
Distribution vector 153
Dominant eigenvalue 111 152 169
Dot product 1
Doubly stochastic matrix 153
Eigenvalue 60
Eigenvalue rank number 82
Eigenvalue, bounds on 112
Eigenvalue, by numerical methods 169—170 181
Eigenvalue, by QR algorithm 181
Eigenvalue, dominant 111 152 169
Eigenvalue, for a unitary matrix 136
Eigenvalue, of a circulant matrix 160
Eigenvalue, of a Hermitian matrix 119
Eigenvalue, of a nonnegative matrix 152
Eigenvalue, of a positive definite matrix 128
Eigenvalue, of a positive matrix 152
Eigenvalue, of a singular matrix 60
Eigenvalue, of a transpose 60
Eigenvalue, of a triangular matrix 60
Eigenvalue, of an elementary reflector 143
Eigenvalue, of an inverse 60
Eigenvalue, of f(A) 93
Eigenvalue, of similar matrices 91
Eigenvector 60
Eigenvector, by numerical methods 169 170
Eigenvector, generalized 82
Eigenvector, maximal linearly independent set of 61
Eigenvector, of a circulant matrix 160
Eigenvector, of a Hermitian matrix 119
Eigenvector, of a nonnegative matrix 152 153
Eigenvector, of a normal matrix 119
Eigenvector, of a real symmetric matrix 119
Elementary column operations 3
Elementary matrix 24
Elementary matrix, determinant of 43
Elementary matrix, inverse of 34
Elementary reflector 137
Elementary row operations 3
Equality of matrices 1
Ergodic matrix 153
Euclidean inner product 103 120
Euclidean matrix norm 111
Euclidean vector norm 110 116 194
Expansion by cofactors 42
Exponential of a matrix 71
Finite Markov chain 153
Frobenius norm 111 116
Function of a matrix 71 92
Function of a matrix, eigenvalues of 93
Function of a matrix, Hermitian 127
Gauss — Jordan elimination 21
Gaussian elimination 12
Generalized eigenvector 82
Generalized inverse 192
Gerschgorin’s theorem 170
Given’s method 167
Gram — Schmidt orthogonalization process 104 121
Gram — Schmidt orthogonalization process, modified 181
Hermitian congruent matrices 145
Hermitian matrix 119
Hermitian matrix, eigenvalues of 119
Hermitian matrix, for a quadratic form 144
Hermitian matrix, functions of 127
Hermitian matrix, generalized inverse of 192
Hermitian matrix, positive definite 128
Hermitian matrix, Rayleigh’s quotient 146
Hermitian transpose matrix 119
Hermitian transpose matrix, for a unitary matrix 136
Hessenberg form 161 182
Homogeneous linear equations 12 23 55 59
Householder transformation 137
Idempotent matrix 33 202
Identity matrix 24 118
Inconsistent simultaneous equations 11
INDEX 145
Index, of a nilpotent matrix 32
Induced norm 111
Inertia matrix 145
Infinite series of matrices 71
Inner product 103 136 144
Integral of a matrix 72
Inverse 34 44
Inverse power method 170
Inverse power method, modified 180
Inverse, and congruence 151
Inverse, by row reduction 34
Inverse, determinant of 43
| Inverse, eigenvalue of 60
Inverse, of a circulant matrix 160
Inverse, of a unitary matrix 136
Inverse, of an elementary matrix 34
Inverse, of an orthogonal matrix 136
Invertible 34
Irreducible matrix 152
Jordan block 91
Jordan canonical form 91
Kronecker product 165
Least squares 194 201
Left eigenvector 67 154
Length, of a chain 82
Length, of a vector 110 143
linear combination 52
Linear combination, of eigenvectors 66
Linear dependence 52 53
Linear equations see “Simultaneous linear equations”
Linear independence 52 53
Linear independence of a chain 85
Linear independence of eigenvectors 61
Lower triangular matrix 24
Lower triangular matrix, determinant of 42
Lower triangular matrix, eigenvalues of 60
Lower triangular matrix, inverse of 32
LU decomposition 24 40 51
LU decomposition, Cholesky decomposition 129
LU decomposition, Crout’s reduction 25 161
Magnitude of a vector 110
Main diagonal 24
Markov chain 153
Matrix 1
Matrix equation AX + XB = C 73
Matrix equation AX + XB = C, AX = B see “Simultaneous linear equations”
Matrix equation AX + XB = C, AXB = C 166
Matrix norm 110
Maximal set of linearly independent vectors 57
Maximal set of linearly independent vectors, eigenvectors 61
Minimum polynomial 83
Minor 42
Minor, principal 128
Modal matrix 91
Modified Gram — Schmidt process 174
Moore — Penrose inverse 192
Multiplication of matrices 2 6
Negative definite matrix 128
Negative semidefinite matrix 128
Nilpotent 32
Nonhomogeneous linear equations 12 (see also “Simultaneous linear equations”)
Nonnegative matrix 152
Nonsingular matrix 34
Nontrivial solutions 12
Norm 110
Normal equations 201
Normal matrix 119 136
Normal matrix, similar to a diagonal matrix 137
Normalized vector 110
Null space 59
Numerical methods for eigenvalues 169 170 181
Order 1
Orthogonal matrix 136 137
Orthogonal vectors 103
Orthonormal vectors 110 119 136
Partial pivoting 13 (see also “Pivoting strategies”)
Partitioned matrix 5
Partitioned matrix, determinant of 43
Permutation matrix 152
Perpendicular see “Orthogonal vectors”
Perron — Frobenius theorem 152
Perron’s theorem 152
Pivot 3
Pivotal condensation 43
Pivoting strategies 12 18 19 25 35
Polar decomposition 202
Positive definite matrix 128
Positive matrix 152
Positive semidefinite matrix 128
Power method 169
Power method, modified 174
Powers of a matrix 26
Primitive matrix 153
Principal diagonal 24
Principal minor 128
Pseudoinverse 192
QR algorithm 181
QR decomposition 181 201
Quadratic form 144
Rank number of an eigenvalue 82
Rank, of a generalized eigenvector 82
Rank, of a matrix 3 12 43 53 144 192
Rayleigh’s principle 146
Rayleigh’s quotient 146 174 180
Real matrix 1
Real symmetric matrix 119 161 174
Real symmetric matrix for a quadratic form 144
Reducible matrix 152
Regular matrix 153
Right eigenvector 60 (see also “Eigenvector”)
Rotation matrix 143 167
Row rank 3 53
Row vector 1
Row-echelon form 2
Row-echelon form, reduction to 3
scalar 1
Scalar multiplication 2
Scaled pivoting 18
Schur decomposition 136
Schwarz inequality 103
Self-adjoint matrix 120
Sequences of matrices 71
Series of matrices 71
Shifted inverse power method 170
Shifted inverse power method, modified 180
Shifted QR algorithm 182
Signature 145
Similar matrices 91 123
Similarity transformation 137
Similarity transformation, QR algorithm 188
Simultaneous linear equations 11 25 35
Simultaneous linear equations, homogeneous 12 55 59
Simultaneous linear equations, matrix form 11 (see also “Solutions of simultaneous linear equations”)
Singular matrix 34
Singular matrix, eigenvalues of 60
Singular value 193
Singular value decomposition 193
Skew-Hermitian matrix 127
Skew-symmetric matrix 127
Solutions of simultaneous linear equations 11
Solutions of simultaneous linear equations, by Gauss — Jordan elimination 21
Solutions of simultaneous linear equations, by Gaussian elimination 12
Solutions of simultaneous linear equations, by inversion 35
Solutions of simultaneous linear equations, by LU decomposition 25
Solutions of simultaneous linear equations, least squares 194
Solutions of simultaneous linear equations, theory of 12 23 55
Solutions of simultaneous linear equations, trivial 12
Spectral norm 111 141
Spectral radius 111 118
Spectral radius, for nonnegative matrices 152
Square matrix 24
Square root of a matrix 128
Stochastic matrix 153 158
Subdiagonal 24
Subtraction of matrices 2
Superdiagonal 24
Sylvester’s law of inertia 145
Symmetric matrix see “Real symmetric matrix”
Toeplitz matrix 160 161
Trace 60 70 91 172
TRANSPOSE 1
Transpose, determinant of 43
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