Harville D.A. — Matrix Algebra: Exercises and Solutions :: Электронная библиотека попечительского совета мехмата МГУ

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Harville D.A. — Matrix Algebra: Exercises and Solutions

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Название: Matrix Algebra: Exercises and Solutions

Автор: Harville D.A.

Аннотация:

This book contains over 300 exercises and solutions covering a wide variety of topics in matrix algebra. They can be used for independent study or in creating a challenging and stimulating environment that encourages active engagement in the learning process. Thus, the book can be of value to both teachers and students. The requisite background is some previous exposure to matrix algebra of the kind obtained in a first course. The exercises are those from an earlier book by the same author entitled Matrix Algebra From a Statistician's Perspective (ISBN 0-387-94978-X). They have been restated (as necessary) to stand alone, and the book includes extensive and detailed summaries of all relevant terminology and notation. The coverage includes topics of special interest and relevance in statistics and related disciplines, as well as standard topics.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 2001

Количество страниц: 312

Добавлена в каталог: 11.04.2008

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Предметный указатель

Adjoint matrix      72 75 232
Adjoint matrix, determinant of      see Determinant
Adjoint matrix, differentiation of      see Differentiation
Adjoint matrix, eigenvalues of      see Eigenvalue(s)
Adjoint matrix, eigenvectors of      see Eigenvector(s)
Adjoint matrix, of a product      77
Algebraic multiplicity      236
Algebraic multiplicity, of zero      235 236
Basis      15
Basis, orthonormal      22 24 31
Bilinear form      79
Binet - Cauchy formula      77
Cayley - Hamilton theorem      232 234
Characteristic polynomial      232 233 243
Characteristic polynomial, of an orthogonal matrix      242
Cofactor matrix      72
Cofactor(s)      74
Cofactor(s), expansion by      see Determinant
Column space(s)      13 15 31 55 167
Column space(s), essential disjointness of      see Essential disjointness
Column space(s), intersection of      161
Column space(s), of a product      27
Column space(s), of a sum      198
Column space(s), orthogonal complement of      see Orthogonal complement
Column space(s), sum of      161
Column space(s), union of      161
Decomposition, $\mathbf{U}'\mathbf{DU}$       87
Decomposition, Cholesky      90 91
Decomposition, LDU      87-91 98 144
Decomposition, of a nonnegative definite matrix      86
Decomposition, of a symmetric matrix      30 83
Decomposition, QR      24 90 91
Decomposition, singular value      245 246
Decomposition, spectral      239 241
Determinant      71 87
Determinant, differentiation of      see Differentiation
Determinant, effect of elementary row or column operations on      70
Determinant, expansion by cofactors      71 74
Determinant, of $\mathbf{R}+\mathbf{STU}$       179 180
Determinant, of a partitioned matrix      71 76
Determinant, of a positive definite matrix      100
Determinant, of a product      see Binet-Cauchy formula
Determinant, of an adjoint matrix      72
Determinant, of an inverse matrix      71
Determinant, of Vandermonde matrix      77
Diagonalization      238
Diagonalization, of a transposed matrix      237
Diagonalization, of an inverse matrix      237
Diagonalization, simultaneous      246
Differentiation, chain rule for      122-124
Differentiation, of a determinant      124
Differentiation, of a Kronecker product      158
Differentiation, of a log of a determinant      125-128 131 132
Differentiation, of a power of a determinant      124
Differentiation, of a power of a function      115
Differentiation, of a product of a scalar and a vector      116
Differentiation, of a projection matrix      135 136
Differentiation, of a trace of a power      119 120
Differentiation, of a trace of a product      116 117 120
Differentiation, of a vec of a Kronecker product      158
Differentiation, of a vec of a matrix power      157
Differentiation, of an adjoint matrix      129
Differentiation, of an idempotent matrix      115
Differentiation, of an inverse matrix      130-132
Differentiation, with respect to a matrix or symmetric matrix      122 125 126 128 130-132
Distance      22
Duplication matrix      154-156
Duplication matrix, left inverse of      154-156
Eigenvalue(s)      231 243
Eigenvalue(s), algebraic multiplicity of      see Algebraic multiplicity
Eigenvalue(s), geometric multiplicity of      see Geometric multiplicity
Eigenvalue(s), not necessarily distinct      238 240
Eigenvalue(s), of a skew-symmetric matrix      231
Eigenvalue(s), of an adjoint matrix      242
Eigenvalue(s), of an idempotent matrix      242
Eigenvalue(s), of an orthogonal matrix      242
Eigenvalue(s), of Moore - Penrose inverse      241
Eigenvectors)      243
Eigenvectors), linear combination of      237
Eigenvectors), of an adjoint matrix      242
Elimination matrix      154
Essential disjointness (of subspaces)      252
Essential disjointness (of subspaces), as applied to row and column spaces      168 198
Function (continuously differentiable)      113
Generalized eigenvalue problem      249 250
Generalized inverse      35 36 91 193 200
Generalized inverse, alternative characterizations for      35 37
Generalized inverse, existence of      35
Generalized inverse, minimum norm      223
Generalized inverse, nonnegative definite      87
Generalized inverse, of $\mathbf{A}'\mathbf{A}$       36
Generalized inverse, of $\mathbf{R}+\mathbf{STU}$       185 186
Generalized inverse, of a block-diagonal matrix      39 45
Generalized inverse, of a partitioned matrix      39 41 42 45 46 52 97 169 170 222
Generalized inverse, of a product      51 106
Generalized inverse, of a scalar multiple      38
Generalized inverse, of a Schur complement      46
Generalized inverse, of a submatrix      45 46
Generalized inverse, reflexive      222
Geometric multiplicity of one      242
Geometric multiplicity of zero      236
Gram - Schmidt orthogonalization      22-24 66
Gram matrix      81
Gramian      81
Hadamard product      95
Hessian matrix      114
Index of inertia      83
Inner product      102 105 147 251
Inner product, quasi      105
Inverse      29 30 74 234
Inverse, determinant of      see Determinant
Inverse, diagonalization of      see Diagonalization
Inverse, differentiation of      see Differentiation
Inverse, of $\mathbf{R}+\mathbf{STU}$       184
Inverse, of a 2 $\times$ 2 matrix      73
Inverse, of a block-triangular matrix      31
Inverse, of a positive definite matrix      82 182
Inverse, of a sum or difference      184 187 188 191
Kronecker product      140 156
Kronecker product, differentiation of      see Differentiation
Kronecker product, generalized inverse of      141
Kronecker product, involving a diagonal matrix      156
Kronecker product, involving a partitioned matrix      143
Kronecker product, involving a sum (or sums)      139
Kronecker product, involving a triangular matrix      144 156
Kronecker product, involving a vector      140 151 154
Kronecker product, LDU decomposition of      144
Kronecker product, nonnegative definite      142
Kronecker product, norm of      143
Kronecker product, of idempotent matrices      140
Kronecker product, of orthogonal matrices      140
Kronecker product, of projection matrices      141
Kronecker product, positive definite      142
Kronecker product, projection matrix for column space of      see Projection matrix
Kronecker product, vec of      see vec
Left inverse      29
Left inverse, of duplication matrix      see Duplication matrix
Left space(s)      13
Left space(s), basis for      see basis
Left space(s), isomorphic      260
Left space(s), subspace(s) of      see subspace(s)
Linear dependence      11 12
Linear independence      11 12 145
Linear system(s), absorption in      211
Linear system(s), augmented      60
Linear system(s), consistent      58
Linear system(s), Cramer's rule for      75
Linear system(s), equivalence of      58 157 210
Linear system(s), homogeneous      55
Linear system(s), inconsistent      58

Linear system(s), invariance to choice of solution of      60
Linear system(s), linear combination of solutions of      55 56
Linear system(s), nonhomogeneous      56
Linear system(s), of the form $\mathbf{X}'\mathbf{XB} = \mathbf{X}'$       65
Linear system(s), solution set of      56 58
Linear system(s), solution to      57 75
Linear transformation(s)      252
Linear transformation(s), dual      261 263
Linear transformation(s), identity      258 261
Linear transformation(s), matrix representation of      255 256 258 260 261 263
Linear transformation(s), null space of      260 264
Linear transformation(s), one to one      251
Linear transformation(s), product of      253 254 262
Linear transformation(s), range space of      264
Linear transformation(s), rank of      264
Linear transformation(s), restriction of      251
Linear transformation(s), scalar multiple of      253 262
Linear transformation(s), sum of      253 262
Linear transformation(s), zero      261
Matrix (or matrices), commutativity of      3
Matrix (or matrices), congruence of      83
Matrix (or matrices), diagonal      7
Matrix (or matrices), diagonally dominant      98
Matrix (or matrices), difference between      3
Matrix (or matrices), idempotent      49 50 82 115 140 146 189-192 195 226
Matrix (or matrices), invertible      29
Matrix (or matrices), involutory      29 50
Matrix (or matrices), negative definite      80 83 142
Matrix (or matrices), negative semidefinite      80
Matrix (or matrices), nonnegative definite      84 90 93 102 142 180 181 188-190 195 198 250
Matrix (or matrices), nonnegative definite, 2 $\times$ 2      102
Matrix (or matrices), nonnegative definite, partitioned      96 97
Matrix (or matrices), nonnegative definite, sum of      93
Matrix (or matrices), nonpositive definite      80 142
Matrix (or matrices), nonsingular      30 98 135
Matrix (or matrices), nonsymmetric      13
Matrix (or matrices), of the form $\mathbf{V}+\mathbf{XUX}'$       209 212
Matrix (or matrices), orthogonal      30 31 49 140 146 221
Matrix (or matrices), permutation      149
Matrix (or matrices), positive definite      80 83 84 98 101 105 135 142 191 250
Matrix (or matrices), positive definite, 2 $\times$ 2      101
Matrix (or matrices), positive definite, product of      94
Matrix (or matrices), positive semidefinite      80 90
Matrix (or matrices), positive semidefinite, nonsingular      80
Matrix (or matrices), positive semidefinite, partitioned      96
Matrix (or matrices), power of      4
Matrix (or matrices), product of      1-3 27
Matrix (or matrices), scalar multiple of      1
Matrix (or matrices), similarity of      see similarity
Matrix (or matrices), singular      72
Matrix (or matrices), skew-symmetric      92 93 183
Matrix (or matrices), submatrix of      see submatrix
Matrix (or matrices), sum of      1-3
Matrix (or matrices), symmetric      3 7 20 36 72
Matrix (or matrices), transpose of      4 9
Matrix (or matrices), triangular      31
Matrix (or matrices), triangular, lower      4 144
Matrix (or matrices), triangular, upper      4 7 8 13 79 144
Minimization (of a 2nd-degree polynomial)      209
Minimization (of a 2nd-degree polynomial), subject to linear constraints      214 216-218 225 248
Minimization (of a 2nd-degree polynomial), transformation from constrained to unconstrained      214 227
Moore - Penrose conditions      223
Moore - Penrose inverse      221 223 225
Moore - Penrose inverse, eigenvalues of      see Eigenvalue(s)
Moore - Penrose inverse, of a nonnegative definite matrix      228
Moore - Penrose inverse, of a product      221 225
Moore - Penrose inverse, of a sum      224
Moore - Penrose inverse, of a symmetric nonnegative definite matrix      226
Neighborhood      113
Norm      21
Norm, limit of      187
Norm, quasi      105
Norm, usual      143
Normal equations      65
Null space      112
Orthogonal complement      177
Orthogonal complement, dimension of      112
Orthogonal complement, of a column space      112
Orthogonal complement, of a sum      162
Orthogonal complement, of an intersection      162
Orthogonal complement, projection on      67 112
Orthogonality of 2 subspaces      63 162 177
Orthogonality of a matrix and a subspace      63 64 162
Orthogonality of a vector and a subspace      107
Partitioned matrix (or matrices), block-triangular      8
Partitioned matrix (or matrices), determinant of      see Determinant
Partitioned matrix (or matrices), generalized inverse of      see Generalized inverse
Partitioned matrix (or matrices), in a Kronecker product      see Kronecker product
Partitioned matrix (or matrices), nonnegative definite      see Matrix (or matrices) nonnegative
Partitioned matrix (or matrices), positive semidefinite      see Matrix (or matrices) positive
Partitioned matrix (or matrices), product of      9
Partitioned matrix (or matrices), rank of      see Rank
Partitioned matrix (or matrices), row space of      see Row space(s)
Partitioned matrix (or matrices), Schur complement in      see Schur complement
Partitioned matrix (or matrices), transpose of      9
Positive or negative pair (of matrix elements)      69
Projection      64
Projection matrix      66 108-110 213
Projection matrix, differentiation of      see Differentiation
Projection matrix, for column space of a Kronecker projection matrix, product      141
Projection matrix, for one subspace along another      173 175 193 212
Projection matrix, in a Kronecker product      see Kronecker product
Projection, along a subspace      173 177
Projection, of a column vector      65 106 111
Projection, on an orthogonal complement      see Orthogonal complement
Quadratic form (matrix of)      79
Rank      15 17 82
Rank, additivity of      191 192 195 199 203
Rank, full column      30
Rank, full row      30
Rank, of $\mathbf{R}+\mathbf{STU}$       196 206
Rank, of a difference      200
Rank, of a partitioned matrix      17 51 53 97 167 168 171 204
Rank, of a product      16 27 172
Rank, of a sum      197 198
Rank, of a triangular matrix      31
Rank, subtractivity of      200
Right inverse      29 174
Row space(s)      13
Row space(s), of a partitioned matrix      17
Row space(s), of a product      27
Row space(s), of a sum      198
Schur complement      42 46 98
Schur complement, generalized inverse of      see Generalized inverse
Schwarz inequality      21
Set, interior point of      114
Set, open      113 135
Set, span of      14 15 163
Similarity      234 235
Similarity, to an idempotent matrix      234
Submatrix, generalized inverse of      see Generalized inverse
Submatrix, principal      7 98
Submatrix, transpose of      7
Subspace(s)      14
Subspace(s), direct sum of      261
Subspace(s), essential disjointness of      see Essential disjointness
Subspace(s), independence of      164 173 177 203
Subspace(s), intersection of      163
Subspace(s), invariant      231 261
Subspace(s), orthogonality of      see Orthogonality
Subspace(s), projection along      see Projection
Subspace(s), projection matrix for      see Projection matrix
Subspace(s), sum of      161-164 170
Subspace(s), union of      161
Sweep operation      43
Trace, differentiation of      see Differentiation
Trace, of a product      19 94 146
Trace, of a sum      96
Transformation(s), inverse of      254 255
Transformation(s), invertible      254 255

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