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| Graybill F.A. — Matrices with Applications in Statistics |
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| Предметный указатель |
Addition of matrices 4
Analysis of variance 3
Approximate solutions of linear equations Sec. 7.4
Approximate solutions of linear equations, with constraints 170
Asymptotic equivalence of matrices 101
Band matrices 282 283
Basis 31
Basis, normal orthogonal 34
Basis, orthogonal 34
Centrosymmetric matrix see Cross-symmetric matrix
Characteristic equations 42
Characteristic polynomial 92
Characteristic roots 42 47 48 98 404
Characteristic roots, bounds on 409
Characteristic roots, of correlation matrix 215
Characteristic roots, of patterned matrices 186 Sec.
Circulants 101 Sec.
Circulants, determinant of 240 241
Circulants, generalized inverse 249
Circulants, inverse 237 242 243
Circulants, product 236 242 244
Circulants, regular 235-241
Circulants, roots 239 246 247
Circulants, sum of 235 242 244
Circulants, symmetric 242 243
Circulants, symmetric regular 241 242
Column space 87 92
Commutation matrix Sec. 9.3
Commutation matrix , direct product 318
Conditional inverse Sec. 6.6 117 118 145-148 172-176 180 181 416 417
Conditional inverse, definition 130
Conditional inverse, rank of 134
Conditional inverse, use in linear equations Sec. 7.3
Conditional inverse, use of Hermite matrices 130
Convergence of matrices 95ff.
Correlation matrix Sec. 8.7
Cosine, direction 62
Covariance matrix 1 188 189 200 201 335 Sec.
Cross-symmetric matrix 287 288
Cross-symmetric matrix, skew 44
Derivative Sec. 10.8
Derivative, of a determinant 355 356
Derivative, of a linear function 352
Derivative, of a matrix 353
Derivative, of a quadratic form 352
Derivative, of inverse 357
Derivative, with respect to a matrix 353
Derivative, with respect to a vector 351
Determinant 7 46 413;
Determinant, of circulants Sec. 8.10
Determinant, of Fourier matrices 272
Determinant, of partitioned matrices 184
Determinant, of patterned matrices Sec. 8.4
Determinant, of the product of matrices 9
Determinant, of Toeplitz matrices 285 286
Determinant, of Vandermonde matrices 266
Determinant, properties of 8
Diagonal matrices 5
Direct product of matrices Sec. 8.8
Direct product of matrices, characteristic roots of 226 227
Direct product of matrices, determinant of 224
Direct product of matrices, inverse of 223
Direct product of matrices, of positive definite matrices 227
Direct product of matrices, product of 225
Direct product of matrices, trace of 301
Direct product of matrices, transpose of 221
Direct product of matrices, use with commutation matrices Sec. 9.3
Direct product of matrices, use with the vector of a matrix Sec. 9.2
Direct sum of matrices 228 229
Distance 54
Distance, point to line 65
Dominant diagonal matrices Sec. 8.11
Dominant diagonal matrices, characteristic roots 261
Dominant diagonal matrices, determinants 258
Dominant diagonal matrices, for solving matrix inequations 265
Dominant diagonal matrices, nonsingular matrices 251-256
Dominant diagonal matrices, reducible and irreducible matrices 263
Elementary operations on matrices 11
Equations, set of linear Chap. 7
Equations, set of linear, c-inverse Sec. 7.3
Equations, set of linear, g-inverse Sec. 6.3 Sec.
Equations, set of linear, L-inverse Sec. 7.6
Euclidean norm 94
Euclidean space 54
Euclidean space, definition 53
Euclidean space, distance 54
Expected value 1 338
Expected value, and direct product Sec. 10.9
Expected value, of a matrix 339
Expected value, of a quadratic form 340 341 Sjc.
Expected value, of powers of quadratic forms Sec. 10.9
Expected value, of products of quadratic forms Sec. 10.9
Expected value, Wishart matrices Sec. 10.9
Experimental design matrix 194 195 230
Exponential distribution 188
Fourier matrix Sec. 8.12
Functions of matrices 92 Sec.
Functions of matrices, norm 92
Functions of matrices, polynomial 92
Functions of matrices, sequences of matrices 94ff.
Generalized inverse Sec. 6.2 172-176 180 181 416 417 440
Generalized inverse, computing Sec. 6.5 145-148
Generalized inverse, definition 106
Generalized inverse, of circulants 249
Generalized inverse, of diagonal matrix 108
Generalized inverse, of nonsingular matrix 108
Generalized inverse, of some special matrices Sec. 6.4
Generalized inverse, of symmetric idempotent matrices 111
Generalized inverse, of symmetric matrix 108
Generalized inverse, rank of 108
Generalized inverse, use in best approximate solution of equations 159
Generalized inverse, use in solution of linear equations Sec. 6.3 152 153
Hadamard matrix Sec. 8.14
Hermite form of matrices 130 Sec.
Hermite form of matrices, use in computing c-inverse 132
Idempotent matrices 3 Sec. 433 439
Independent vectors 27
Infinite sum of matrices 96ff.
Inverse 198 199 200; General¬ized Least
Inverse positive matrices Chap. 11
Irreducible matrix 263
Irreducible matrix and dominant diagonal matrix 264
Latin square matrix 194
Leading principal minor 9
Least squares 2
Least squares inverse Sec. 7.6
Least squares inverse, computation of 169
Least squares inverse, definition of 164
Least squares inverse, use in solution of linear equations 168
Least squares inverse, with constraints 170-176
Least squares, weighted 177
linear combination 27
Linear dependence 27
Linear equations, approximate solution to Sec. 7.4
Linear equations, best approximate solution 157 159 168
Linear equations, consistency conditions for 150-153
Linear equations, number of solutions Sec. 7.3
Linear equations, solution of 151-155
Linear equations, two systems 176
Linear equations, use of c-inverse 151-155
Linear equations, use of g-inverse 151-155. 159
Linear equations, use of L-inverse 168
Linear equations, with constraints 170ff.
Linear independence Sec. 2.4
Linear model 1 2 92
Linear transformations 39
Linear transformations, of vector space 41
Lines, angle between two 59 70
Lines, direction angles of 61
Lines, in  56
| Lines, intersection of 57 70
Lines, parallel 57
Lines, projection of 70
lines, segments 60
Lower triangular matrix see Triangular matrix
M-matrices Sec. 11.4
Matrices type 2 198
Matrices, algebra of 4 5
Matrices, characteristic equation of 42
Matrices, characteristic polynomial of 42
Matrices, characteristic roots of 42 47 48
Matrices, characteristic vectors of 42
Matrices, circulants Sec. 8.10
Matrices, column space of 87
Matrices, congruent 16
Matrices, correlation Sec. 8.7
Matrices, covariance 335 Sec.
Matrices, diagonal 5
Matrices, direct product of Sec. 8.8
Matrices, direct sum of Sec. 8.8
Matrices, elementary 12
Matrices, elementary transformations 11
Matrices, Hermite form of see Hermite form
Matrices, idempotent see Idempotent matrices
Matrices, inverse see Inverse
Matrices, non-negative Sec. 12.1
Matrices, null space 87
Matrices, orthogonal see Orthogonal
Matrices, P-matrices Sec. 11.2
Matrices, partitioned Sec. 8.2
Matrices, positive definite 208 Sec. 438;
Matrices, positive semidefinite Sec. 12.2
Matrices, rank 10
Matrices, reducible 263
Matrices, similar 45
Matrices, skew symmetric 415
Matrices, symmetric 7 47 48
Matrices, trace of see Trace
Matrices, transformation of random variables Sec. 10.2
Matrices, transpose of 6
Matrices, triangular see Triangular
Matrices, tripotent matrices see Tripotent matrices
Matrices, with non-positive off-diagonal elements see Z-matrices
Matrices, with positive elements 374
Matrices, with positive principal minors see P-matrices M-matrices
Matrices, Z-matrices Sec. 11.3
Maximum likelihood 2
Minimum polynomial 92
Moment generating function 350
Moments of normal distribution Sec. 8.4
Multinomial distribution 189
Multivariate normal distribution Sec. 10.3
n-dimensional geometry 53
n-dimensional geometry, equation of a line 54
n-dimensional geometry, equation of a plane 65
n-dimensional vector space 53
n-tuples, ordered 23
n-tuples, space of 24
Non-negative matrices Sec. 12.2
Norm 93ff.
Normal distribution Sec. 10.3
Normal distribution, covariance Sec. 10.3 Sec.
Normal distribution, mean Sec. 10.4
Normal distribution, variance Sec. 10.4
Normal equations 161
Null matrix 5
Null space 87
Order statistics 188
Orthogonal complement of a subspace 83
Orthogonal matrix 18 48
Orthogonal projections see Projections
Orthogonal set of vectors 35 47
Orthogonal transformations 19
P-matrices Sec. 11.2 Sec.
Parameter space 91
Partitioned matrix Sec. 8.5
Partitioned matrix, characteristic roots Sec. 8.5
Partitioned matrix, determinant of 184 231
Partitioned matrix, generalized inverse of Sec. 7.6
Partitioned matrix, inverse of 184
Patterned matrices Sec. 8.3 Sec.
Patterned matrices, conditional inverse of 449
Patterned matrices, determinant of 184 Sec.
Patterned matrices, generalized inverse of Sec. 7.6
Patterned matrices, inverse of 184 Sec.
Permutation matrices Sec. 8.13
Planes 65 69
Positive definite Sec. 12.2
Positive semidefinite Sec. 12.2
Projections 70 85 Sec.
Quadratic form 14 273 Sec.
Quadratic form, derivative of 352
Quadratic form, expected value Sec. 12.2
Quadratic form, index 16
Quadratic form, matrix of 15
Quadratic form, non-negative Sec. 12.2
Quadratic form, positive definite Sec. 12.2 17
Quadratic form, positive semidefinite 17 Sec.
Quadratic form, real symmetric 16
Quadratic function 15
Random variables Sec. 10.2
Random variables, expected value 338 339 Sec.
Random variables, moments of Sec. 10.2
Random variables, transformations Sec. 10.2
Range space 87
Rank Sec. 1.6; see under each type of matrix
Rank of a matrix 18
Rank of conditional inverse 134
Rank of direct product 227
Rank of the generalized inverse 109
Rectangular distribution 100
Reducible matrix 263
Sequence of matrices 95ff.
Sequence of matrices, limit of 95
Skew-symmetric matrix 444
Space 87 92 column
Spectral norm 94
Statistical applications Sec. 7.5 Sec.
Statistical applications, use of generalized inverse 162 163
Subspaces 25
Subspaces, intersection of 79
Subspaces, orthogonal 83
Subspaces, sum of 80
Symmetric matrix see Matrices symmetric
Systems of linear equations Sec. 6.3
Systems of linear equations, augmented matrix of 151
Systems of linear equations, best approximate solutions Sec. 7.4
Systems of linear equations, consistent 149
Systems of linear equations, least squares solution Sec. 7.4
Systems of linear equations, solution of 152-156
Toeplitz matrix 101 Sec.
Trace of a matrix Sec. 9.1
Trace of a matrix, of non-negative matrices 397 398 399
TRANSPOSE Sec. 1.4
Triangular matrix 46 Sec.
Triangular matrix, characteristic roots of 210 212
Triangular matrix, determinant of 210
Triangular matrix, factoring into 207 208 209 210 212
Triangular matrix, inverse of 209
Triangular matrix, product of 212 213
Triangular matrix, reduction to 211 212 213
Tripotent matrices Sec. 12.4 438
Vandermonde matrix Sec. 8.12
Variance 1
Variance of quadratic forms Sec. 10.9
Vector addition 24
Vector components 23
Vector direction 61
Vector equality 24
Vector of a matrix Sec. 9.2 316 317
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