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Schott J.R. Ч Matrix Analysis for Statistics
Schott J.R. Ч Matrix Analysis for Statistics

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Ќазвание: Matrix Analysis for Statistics

јвтор: Schott J.R.


A complete, self-contained introduction to matrix analysis theory and practice
Matrix methods have evolved from a tool for expressing statistical problems to an indispensable part of the development, understanding, and use of various types of complex statistical analyses. As such, they have become a vital part of any statistical education. Unfortunately, matrix methods are usually treated piecemeal in courses on everything from regression analysis to stochastic processes. Matrix Analysis for Statistics offers a unique view of matrix analysis theory and methods as a whole.
Professor James R. Schott provides in-depth, step-by-step coverage of the most common matrix methods now used in statistical applications, including eigenvalues and eigenvectors, the Moore-Penrose inverse, matrix differentiation, the distribution of quadratic forms, and more. The subject matter is presented in a theorem/proof format, and every effort has been made to ease the transition from one topic to another. Proofs are easy to follow, and the author carefully justifies every step. Accessible even for readers with a cursory background in statistics, the text uses examples that are familiar and easy to understand. Other key features that make this the ideal introduction to matrix analysis theory and practice include:
- Self-contained chapters for flexibility in topic choice.
- Extensive examples and chapter-end practice exercises.
- Optional sections for mathematically advanced readers.

язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1996

 оличество страниц: 431

ƒобавлена в каталог: 21.10.2010

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
ѕредметный указатель
Accumulation point      70
Adjoint      8
Analysis of variance      120 see "Two-way
Bartlett adjustment      406
Basis      41Ч43
Basis, orthonormal      48Ч52
Bilinear form      15
Block diagonal matrix      12
Boundary point      72
canonical variate analysis      107 154Ч155 406Ч407
Cauchy Ч Schwarz inequality      35
Cayley Ч Hamilton theorem      93
Chain rule      324 327
Characteristic equation      85
Characteristic root      84 see
Characteristic vector      84 see
Chi-squared distribution and Moore Ч Penrose inverse      179Ч180
Chi-squared distribution and quadratic forms      378Ч384
Chi-squared distribution, central      20Ч21
Chi-squared distribution, noncentra]      21
Cholesky decomposition      139
Circulant matrix      300Ч304
Closure      70
Cochran's theorem      374Ч378
Cofactor      5 8
Cofactor, expansion formula for determinant      5Ч6
Column space      43
Commutation matrix      276Ч283
Commutation matrix, eigenvalues      281
Commutation matrix, eigenvectors      317
Complex matrix      16Ч18
Concave function      see "Convex function"
Consistent equations      210Ч213
Consistent estimator      189Ч190
Continuity of determinant      188
Continuity of eigenvalues      103
Continuity of inverse matrix      188
Continuity of Moore Ч Penrose inverse      189
Convex combination      70
Convex function      349Ч353
Convex function, absolute maximum      352
Convex hull      70
Convex set      70Ч74
Correlation coefficient      24
Correlation coefficient, maximum squared      368
Correlation matrix      24
Correlation matrix, nonnegative definite      24
Correlation matrix, sample      25
Courant Ч Fischer min-max theorem      108Ч110
Covariance      22Ч23
Covariance matrix      23
Covariance matrix, nonnegative definite      23
Covariance matrix, sample      25
Covariance of quadratic forms      391 394
Decomposition, Cholesky      139
Decomposition, Jordan      147Ч149
Decomposition, LU      169
Decomposition, QR      140
Decomposition, Schur      149Ч153
Decomposition, singular value      131Ч138
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 336Ч337
Derivative of Moore Ч Penrose inverse      333Ч334 336Ч337
Derivative of patterned matrices      335Ч337
Derivative of trace      332
Derivative of vector function      327
Derivative, partial      325
Derivative, second-order partial      326
Determinant      5Ч8
Determinant and eigenvalues      90
Determinant of partitioned matrix      249Ч250
Determinant, continuity of      188
Determinant, derivative of      332 336
Determinant, expansion formula for      5Ч6
Diagonal matrix      2
Diagonalization      92 144Ч147
Diagonalization, simultaneous      118 154Ч157
Differential      324 325
Differential of determinant      332
Differential of eigenvalue      343
Differential of eigenvector      343
Differential of inverse      333 336Ч337
Differential of matrix function      328
Differential of Moore Ч Penrose inverse      334Ч335 336Ч337
Differential of trace      332
Differential of vector function      327
Differential, second      326
Dimension of vector space      41
Direct sum of matrices      260Ч261
Discriminant analysis      37
Distance function      36
Distance function, euclidean      36 50 62Ч63 141
Distance function, Mahalanobis      37 63 141
Distance in the metric of      37
Duplication matrix      238Ч285
Eigenprojection      98
Eigenprojection, continuity of      103
Eigenspace      87 146
Eigenvalue      84
Eigenvalue and rank      92 99 146Ч147 153
Eigenvalue in the metric of      118
Eigenvalue of idempotent matrix      370Ч371
Eigenvalue of orthogonal matrix      88
Eigenvalue of positive definite matrix      112
Eigenvalue of positive semidefinite matrix      112
Eigenvalue of symmetric matrix      93Ч102
Eigenvalue of transpose product      114Ч115
Eigenvalue of triangular matrix      88
Eigenvalue, asymptotic distribution of      404Ч406
Eigenvalue, continuity of      103
Eigenvalue, derivative of      343
Eigenvalue, distinct      86
Eigenvalue, extremal properties      104Ч110
Eigenvalue, monotonicity      115
Eigenvalue, multiple      86
Eigenvalue, perturbation of      339Ч343
Eigenvalue, simple      86
Eigenvectors      84
Eigenvectors of symmetric matrix      94Ч96
Eigenvectors, asymptotic distribution of      404Ч406
Eigenvectors, common      128 157
Eigenvectors, derivative of      343
Eigenvectors, linear independence of      91
Elementary transformations      13
Elimination matrices      285Ч288
Estimable function      230
Euclidean norm      36 37 158
Euclidean space      36
Euler's formula      17
Expected value      19
Expected value of quadratic form      390Ч398
F distribution      21Ч22
Fourier matrix      303Ч304
Gauss Ч Seidel method      236
Generalized inverse      190Ч196 see
Generalized inverse, computation of      200Ч203
Generalized inverse, properties      193
Gradient      237
Gram Ч Schmidt orthonormalization      48 54Ч55
Hadamard inequality      270
Hadamard matrix      305Ч307
Hadamard matrix, normalized      306
Hadamard product      266Ч276
Hadamard product as a Kronecker product      267
Hadamard product, eigenvalues of      274Ч276
Hadamard product, rank of      267
Hermite form      200
Hermitian matrix      18
Hessian matrix      326
Homogeneous system of equations      219Ч221
Hyperplane      71
Idempotent matrix      3 58Ч59 370Ч374
Idempotent matrix, eigenvalues of      370Ч371
Idempotent matrix, rank of      370Ч371
Idempotent matrix, symmetric      372 373Ч374
Idempotent matrix, trace of      370Ч371
Identity matrix      2
Indefinite matrix      16
Independence (linear)      38Ч40
Independence (stochastic) of quadratic forms      384Ч390
Independence (stochastic) of random variables      22
Inner product      34Ч35
Inner product, Euclidean      35
Interior point      72
Intersection of vector spaces      67
Inverse matrix      8Ч11
Inverse matrix and cofactors      8Ч9
Inverse matrix of a sum      9Ч10
Inverse matrix of partitioned matrix      347
Inverse matrix, continuity of      188
Inverse matrix, derivative of      333 336Ч337
Irreducible matrix      294Ч295
Jacobi method      236
Jacobian matrix      327
Jensen's inequality      352Ч353
Jordan decomposition      147Ч149
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 Ч 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      113Ч114
Least squares and multicollinearity      96Ч98 136
Least squares and solutions to a system of equations      222Ч228 345Ч346
Least squares in less than full rank models      58 228Ч232
Least squares in multiple regression      55Ч58
Least squares in one-way classification model      79Ч80
Least squares in ridge regression      123
Least squares in simple linear regression      50Ч51
Least squares inverse      196Ч197
Least squares inverse, computation of      203Ч204
Least squares with standardized explanatory variables      64Ч65
Least squares, generalized      141Ч142 245
Least squares, ordinary      26Ч28
Least squares, restricted      80Ч81 245
Least squares, weighted      65Ч66
Limit point      70
linear combination      33
Linear dependence      38Ч40
Linear equations      66Ч67
Linear equations and singular value decomposition      233Ч235
Linear equations, consistency of      210Ч213
Linear equations, homogeneous system of      219Ч221
Linear equations, least squares solutions of      222Ч228
Linear equations, linearly independent solutions to      217
Linear equations, solutions to      213Ч219
Linear equations, sparce systems of      235Ч241
Linear equations, sparce systems of, direct methods      235Ч236
Linear equations, sparce systems of, iterative methods      236Ч241
Linear equations, unique solution to      216
Linear independence      38Ч40
Linear model      27
Linear space      33
Linear transformation      60Ч67
LU factorization      169
Mahalanobis distance      37 63 141
Markov chain      298Ч300
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      300Ч304
Matrix, commutation      276Ч283
Matrix, complex      16Ч18
Matrix, correlation      24
Matrix, covariance      23
Matrix, diagonal      2
Matrix, duplication      283Ч285
Matrix, eigenprojection      98
Matrix, elimination      285Ч288
Matrix, Fourier      303Ч304
Matrix, Hadamard      305Ч307
Matrix, hermitian      18
Matrix, Hessian      326
Matrix, idempotent      3 58Ч59 370Ч374
Matrix, identity      2
Matrix, indefinite      16
Matrix, irreducible      294Ч295
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      14Ч15
Matrix, partitioned      11Ч13
Matrix, permutation      15
Matrix, positive      288
Matrix, positive definite      15Ч16
Matrix, positive semidefinite      15Ч16
Matrix, primitive      298
Matrix, projection      52Ч59
Matrix, reducible      294Ч295
Matrix, similar      144
Matrix, skew-symmetric      4
Matrix, square root      16
Matrix, symmetric      4
Matrix, Toeplitz      304Ч305
Matrix, transpose      3
Matrix, triangular      2
Matrix, unitary      18 150
Matrix, Vandermonde      307Ч309
Maximum Likelihood Estimation      347Ч349
Maximum of a concave function      351
Maximum with equality constraints      353Ч360
Maximum, absolute      344
Maximum, conditions for local maximum      345
Maximum, local      344
Mean      19
Mean squared error      163Ч164
Mean vector      22
Mean vector, differences in      106Ч107 116Ч117 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      19Ч20
1 2
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