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Schott J.R. — Matrix Analysis for Statistics
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Íàçâàíèå: Matrix Analysis for StatisticsÀâòîð: Schott J.R. Àííîòàöèÿ: A complete, self-contained introduction to matrix analysis theory and practice
ßçûê: Ðóáðèêà: Ìàòåìàòèêà /Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö ed2k: ed2k stats Ãîä èçäàíèÿ: 1996Êîëè÷åñòâî ñòðàíèö: 431Äîáàâëåíà â êàòàëîã: 21.10.2010Îïåðàöèè: Ïîëîæèòü íà ïîëêó |
Ñêîïèðîâàòü ññûëêó äëÿ ôîðóìà | Ñêîïèðîâàòü ID
Ïðåäìåòíûé óêàçàòåëü
Accumulation point 70 Adjoint 8 Analysis of variance 120 see "Two-way Bartlett adjustment 406 Basis 41—43 Basis, orthonormal 48—52 Bilinear form 15 Block diagonal matrix 12 Boundary point 72 canonical variate analysis 107 154—155 406—407 Cauchy — Schwarz inequality 35 Cayley — Hamilton theorem 93 Chain rule 324 327 Characteristic equation 85 Characteristic root 84 see Characteristic vector 84 see Chi-squared distribution and Moore — Penrose inverse 179—180 Chi-squared distribution and quadratic forms 378—384 Chi-squared distribution, central 20—21 Chi-squared distribution, noncentra] 21 Cholesky decomposition 139 Circulant matrix 300—304 Closure 70 Cochran's theorem 374—378 Cofactor 5 8 Cofactor, expansion formula for determinant 5—6 Column space 43 Commutation matrix 276—283 Commutation matrix, eigenvalues 281 Commutation matrix, eigenvectors 317 Complex matrix 16—18 Concave function see "Convex function" Consistent equations 210—213 Consistent estimator 189—190 Continuity of determinant 188 Continuity of eigenvalues 103 Continuity of inverse matrix 188 Continuity of Moore — Penrose inverse 189 Convex combination 70 Convex function 349—353 Convex function, absolute maximum 352 Convex hull 70 Convex set 70—74 Correlation coefficient 24 Correlation coefficient, maximum squared 368 Correlation matrix 24 Correlation matrix, nonnegative definite 24 Correlation matrix, sample 25 Courant — Fischer min-max theorem 108—110 Covariance 22—23 Covariance matrix 23 Covariance matrix, nonnegative definite 23 Covariance matrix, sample 25 Covariance of quadratic forms 391 394 Decomposition, Cholesky 139 Decomposition, Jordan 147—149 Decomposition, LU 169 Decomposition, QR 140 Decomposition, Schur 149—153 Decomposition, singular value 131—138 Decomposition, spectral 95 98 138 Density function 19 Derivative 323 325 derivative of determinant 332 336 Derivative of eigenvalue 343 derivative of eigenvector 343 Derivative of inverse 333 336—337 Derivative of Moore — Penrose inverse 333—334 336—337 Derivative of patterned matrices 335—337 Derivative of trace 332 Derivative of vector function 327 Derivative, partial 325 Derivative, second-order partial 326 Determinant 5—8 Determinant and eigenvalues 90 Determinant of partitioned matrix 249—250 Determinant, continuity of 188 Determinant, derivative of 332 336 Determinant, expansion formula for 5—6 Diagonal matrix 2 Diagonalization 92 144—147 Diagonalization, simultaneous 118 154—157 Differential 324 325 Differential of determinant 332 Differential of eigenvalue 343 Differential of eigenvector 343 Differential of inverse 333 336—337 Differential of matrix function 328 Differential of Moore — Penrose inverse 334—335 336—337 Differential of trace 332 Differential of vector function 327 Differential, second 326 Dimension of vector space 41 Direct sum of matrices 260—261 Discriminant analysis 37 Distance function 36 Distance function, euclidean 36 50 62—63 141 Distance function, Mahalanobis 37 63 141 Distance in the metric of 37 Duplication matrix 238—285 Eigenprojection 98 Eigenprojection, continuity of 103 Eigenspace 87 146 Eigenvalue 84 Eigenvalue and rank 92 99 146—147 153 Eigenvalue in the metric of 118 Eigenvalue of idempotent matrix 370—371 Eigenvalue of orthogonal matrix 88 Eigenvalue of positive definite matrix 112 Eigenvalue of positive semidefinite matrix 112 Eigenvalue of symmetric matrix 93—102 Eigenvalue of transpose product 114—115 Eigenvalue of triangular matrix 88 Eigenvalue, asymptotic distribution of 404—406 Eigenvalue, continuity of 103 Eigenvalue, derivative of 343 Eigenvalue, distinct 86 Eigenvalue, extremal properties 104—110 Eigenvalue, monotonicity 115 Eigenvalue, multiple 86 Eigenvalue, perturbation of 339—343 Eigenvalue, simple 86 Eigenvectors 84 Eigenvectors of symmetric matrix 94—96 Eigenvectors, asymptotic distribution of 404—406 Eigenvectors, common 128 157 Eigenvectors, derivative of 343 Eigenvectors, linear independence of 91 Elementary transformations 13 Elimination matrices 285—288 Estimable function 230 Euclidean norm 36 37 158 Euclidean space 36 Euler's formula 17 Expected value 19 Expected value of quadratic form 390—398 F distribution 21—22 Fourier matrix 303—304 Gauss — Seidel method 236 Generalized inverse 190—196 see Generalized inverse, computation of 200—203 Generalized inverse, properties 193 Gradient 237 Gram — Schmidt orthonormalization 48 54—55 Hadamard inequality 270 Hadamard matrix 305—307 Hadamard matrix, normalized 306 Hadamard product 266—276 Hadamard product as a Kronecker product 267 Hadamard product, eigenvalues of 274—276 Hadamard product, rank of 267 Hermite form 200 Hermitian matrix 18 Hessian matrix 326 Homogeneous system of equations 219—221 Hyperplane 71 Idempotent matrix 3 58—59 370—374 Idempotent matrix, eigenvalues of 370—371 Idempotent matrix, rank of 370—371 Idempotent matrix, symmetric 372 373—374 Idempotent matrix, trace of 370—371 Identity matrix 2 Indefinite matrix 16 Independence (linear) 38—40 Independence (stochastic) of quadratic forms 384—390 Independence (stochastic) of random variables 22 Inner product 34—35 Inner product, Euclidean 35 Interior point 72 Intersection of vector spaces 67 Inverse matrix 8—11 Inverse matrix and cofactors 8—9 Inverse matrix of a sum 9—10 Inverse matrix of partitioned matrix 347 Inverse matrix, continuity of 188 Inverse matrix, derivative of 333 336—337 Irreducible matrix 294—295 Jacobi method 236 Jacobian matrix 327 Jensen's inequality 352—353 Jordan decomposition 147—149 Kronecker product 253 Kronecker product, determinant of 256 Kronecker product, eigenvalues of 255 Kronecker product, eigenvectors of 312 Kronecker product, inverse of 255 Kronecker product, Moore — Penrose inverse of 255 Kronecker product, rank of 257 Kronecker product, trace of 255 Lagrange function 354 Lagrange multipliers 354 Lanczos vectors 238 latent root 84 see Latent vector 84 see Least squares see also "Regression" Least squares and best linear unbiased estimator 113—114 Least squares and multicollinearity 96—98 136 Least squares and solutions to a system of equations 222—228 345—346 Least squares in less than full rank models 58 228—232 Least squares in multiple regression 55—58 Least squares in one-way classification model 79—80 Least squares in ridge regression 123 Least squares in simple linear regression 50—51 Least squares inverse 196—197 Least squares inverse, computation of 203—204 Least squares with standardized explanatory variables 64—65 Least squares, generalized 141—142 245 Least squares, ordinary 26—28 Least squares, restricted 80—81 245 Least squares, weighted 65—66 Limit point 70 linear combination 33 Linear dependence 38—40 Linear equations 66—67 Linear equations and singular value decomposition 233—235 Linear equations, consistency of 210—213 Linear equations, homogeneous system of 219—221 Linear equations, least squares solutions of 222—228 Linear equations, linearly independent solutions to 217 Linear equations, solutions to 213—219 Linear equations, sparce systems of 235—241 Linear equations, sparce systems of, direct methods 235—236 Linear equations, sparce systems of, iterative methods 236—241 Linear equations, unique solution to 216 Linear independence 38—40 Linear model 27 Linear space 33 Linear transformation 60—67 LU factorization 169 Mahalanobis distance 37 63 141 Markov chain 298—300 Matrix function 327 Matrix norm 158 Matrix norm, Euclidean 158 Matrix norm, maximum column sum 158 Matrix norm, maximum row sum 158 Matrix norm, spectral 158 Matrix, block diagonal 12 Matrix, circulant 300—304 Matrix, commutation 276—283 Matrix, complex 16—18 Matrix, correlation 24 Matrix, covariance 23 Matrix, diagonal 2 Matrix, duplication 283—285 Matrix, eigenprojection 98 Matrix, elimination 285—288 Matrix, Fourier 303—304 Matrix, Hadamard 305—307 Matrix, hermitian 18 Matrix, Hessian 326 Matrix, idempotent 3 58—59 370—374 Matrix, identity 2 Matrix, indefinite 16 Matrix, irreducible 294—295 Matrix, Jacobian 327 Matrix, negative definite 16 Matrix, negative semidefinite 16 Matrix, nilpotent 127 166 Matrix, nonnegative 288 Matrix, nonnegative definite 16 Matrix, nonsingular 8 Matrix, null 2 Matrix, order of 1 Matrix, orthogonal 14—15 Matrix, partitioned 11—13 Matrix, permutation 15 Matrix, positive 288 Matrix, positive definite 15—16 Matrix, positive semidefinite 15—16 Matrix, primitive 298 Matrix, projection 52—59 Matrix, reducible 294—295 Matrix, similar 144 Matrix, skew-symmetric 4 Matrix, square root 16 Matrix, symmetric 4 Matrix, Toeplitz 304—305 Matrix, transpose 3 Matrix, triangular 2 Matrix, unitary 18 150 Matrix, Vandermonde 307—309 Maximum Likelihood Estimation 347—349 Maximum of a concave function 351 Maximum with equality constraints 353—360 Maximum, absolute 344 Maximum, conditions for local maximum 345 Maximum, local 344 Mean 19 Mean squared error 163—164 Mean vector 22 Mean vector, differences in 106—107 116—117 154 Mean vector, sample 25 Mean, sample 25 Minimum see "Maximum" Minor 5 13 Minor, leading principal 311 Modulus of a complex number 17 Moment generating function 20 Moments 19—20
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