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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.

Язык: 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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 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, 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 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 36 Generalized inverse, of 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 184 Inverse, of a 2 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 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 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 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 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 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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