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Mirsky L. — An introduction to linear algebra
Mirsky L. — An introduction to linear algebra



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Название: An introduction to linear algebra

Автор: Mirsky L.

Аннотация:

Rigorous, self-contained coverage of determinants, vectors, matrices and linear equations, quadratic forms, more. Elementary, easily readable account with numerous examples and problems at the end of each chapter. "The straight-forward clarity of the writing is admirable."—American Mathematical Monthly. Bibliography.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1990

Количество страниц: 433

Добавлена в каталог: 31.10.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abelian groups      254
Adjugate, determinant      24-27
Adjugate, determinant, matrix      88-90
Affine classification of quadrics      381-4
Algebraic complement      see Cofactor.
Alternating group      287
Angle of an orthogonal matrix      236
Arrangements      2-5
Augmented matrix      140
Automorphisms of a linoar manifold      125-6 268-9
Axis of rotation      240
Basis, basis theorems      51-53
Basis, change of basis in a linear manifold      111-13
Basis, completion      54
Basis, orthogonal and orthonormal      65-68 155-6
Bilinear, forms      354-6 368-9
Bilinear, operators      353-6
Bilinear, symmetric bilinear operators      361
Biunique correspondence      4
Bondixon’s inequality      210
Bossel’s inequality      70
Bromwich’s inequality      388-9
Canonical form(s), classical      312
Canonical form(s), classical of matrices      290-316
Canonical form(s), classical of quadratic forms      378
Cayley and Hamilton, theorem of      206
Characteristic, equation      195
Characteristic, inequalities for characteristic roots      210-12 309-11 388-90
Characteristic, polynomial      195 197-208
Characteristic, regular characteristic roots      294
Characteristic, roots      195
Characteristic, roots of rational functions of matrices      201-2
Characteristic, vectors      214
Circulant      36
Classical canonical form      312
Classification of quadrics      380-5
Cofactor, of an element      14
Cofactor, of an element of a minor      21
Collineations      123 196 256 380-5
Column, matrix      77
Column, matrix, rank      137
Column, matrix, vector      77
Commuting matrices      81
Complements      64-57
Complements, orthogonal      68-69
Completion of basis      54
Congruence transformations      182-5 266 358-60
Congruent matrices      182
Consistency of systems of linear equations      132 140-1
Convergence, of matrix sequences      327-9
Convergence, of matrix sequences of matrix series      330-2
Cramer’s Rule      134
Critical minor      136
Definite hermitianand quadratic forms      394 398-404
Determinant rank      136
Determinant(s), adjugate      24-27
Determinant(s), axiomatic characterization      189-92
Determinant(s), definition      6
Determinant(s), dominated by diagonal elements      32-33
Determinant(s), expansions by rows and columns      15
Determinant(s), Gram determinant      155
Determinant(s), inequalities      33 212-13 416-22
Determinant(s), multiplication      12-13
Determinant(s), of a matrix      87
Determinant(s), Vandermonde determinant      17-18
Determinantal criteria for value classes      400-7
Diagonal, canonical forms      290-306
Diagonal, matrix      76
Differential, equations      343 8
Differential, operators      126-7 133-4
Differentiation, of determinants      36
Differentiation, of matrices      339
Dimensionality, of a linear manifold      50
Dimensionality, theorem for homogeneous systems      149
Discriminant      18-19
Disposable unknowns      142
Divisors of zero      95-96 153-4
Elementary, divisors      194
Elementary, matrices ($E$-matrices)      170
Elementary, operations ($E$- operations)      168-78
Equivalence, class      187
Equivalence, of bilinear forms      368
Equivalence, of matrices      176
Equivalence, of quadratic forms      376-9
Equivalence, of systems of linear equations      142
Equivalence, relation      186
Equivalence, transformations      177
Equivalence, w.r.t. operators      189 267
Euler, equations of transformation      242
Euler, theorem on rigid motion      246
Exponential function of a matrix      336-41
Field      39
Field, real and complex      40
Finite groups      254 270-1
Fischer’s inequality      420
Forms, bilinear      364-6 368-9
Forms, hermitian      386-8 416
Forms, polarized      362
Forms, quadratic      357-60
Frobenius, determinantal criterion for positive definite forms      400
Frobenius, theorem on non-negative matrices      329
Frobenius, theorem on rank      162
Full linear group      263 268-9
Functions of matrices      336 342-3
Fundamental sets of solutions      151
Generators, of a linear manifold      45
Generators, of a linear manifold of a vector space      43
Gram, determinant and matrix      155
Gram, determinant and matrix, inequality      423
Group(s), Abelian      254
Group(s), alternating      287
Group(s), axioms      252
Group(s), centre      254
Group(s), congruence group      266
Group(s), finite      254 270-1
Group(s), full linear      263 268-9
Group(s), group matrix      273
Group(s), isomorphism      259
Group(s), multiplication table      1
Group(s), of matrices      261-5
Group(s), of permutations      257-8 271-2
Group(s), of singular matrices      272-6
Group(s), order      254
Group(s), orthogonal      263
Group(s), orthogonal similarity group      266
Group(s), projective      263
Group(s), representation by matrices      267-72
Group(s), rotation group      263
Group(s), similarity group      266
Group(s), symmetric      257
Hadamard’s inequality      418-19
Hamilton      see Cayley.
Hermitian, forms      386-8 415
Hermitian, matrices      209 301 304
Hermitian, positive definite forms      394 398-404
Hirsch’s inequality      211
Homogeneous linear equations      27-30 131 148-52
Homogeneous linear equations, dimensionality theorem      149
Homogeneous linear equations, fundamental sots of solutions      151
Idempotent matrix      107
Image space      277
Indefinite hermitian and quadratic forms      394 407
Index laws for matrices      83 93-94
Inequality(ies), for characteristic roots      210-12 309-11 388-90
Inequality(ies), for determinants      33 212-13
Inequality(ies), for positive definite matrices      416-22
Inequality(ies), of Bendixon      210
Inequality(ies), of Bessel      70
Inequality(ies), of Bromwich      388-9
Inequality(ies), of Fischer      420
Inequality(ies), of Gram      423
Inequality(ies), of Hadamard      418-19
Inequality(ies), of Hirsch      211
Inequality(ies), of Minkowski      419-20
Inequality(ies), of Oppenheim      421
Inequality(ies), of Ostrowski      212-13
Inequality(ies), of Schur      309-11 422
Inequality(ies), triangle      64
Inertia, law of, for hermitian forms      387
Inertia, law of, for hermitian forms for quadratic former      377
Infinitesimal rotation      249
Inner product      62
Interpolation formula of Sylvester      221
Invariance      169
Invariant, properties of matrices      122
Invariant, properties of matrices, spaces      277-86
Inverse matrix      91 178-9
Isomorphism, between groups      259
Isomorphism, betweon linear manifolds      68-59 123-4
Jacobi’s theorem      25
Joachimsthal’s equation      360-2
Klein four-group      261
Kronecker delta      20
Lagrango’s method of reduction      371 4
Laplace’s expansion theorem      21
Latent roots      see Characteristic roots.
Latent vectors      see Characteristic vectors.
Leading element      173 (footnote)
Length of a vector      63
Linear manifold (s), automorphism      6 268-9
Linear manifold (s), basis      49-54
Linear manifold (s), change of basis      111-13
Linear manifold (s), definition      44
Linear manifold (s), dimensionality      50
Linear manifold (s), generators      45
Linear manifold (s), isomorphism      58-59 123-4
Linear manifold (s), of matrices      80
Linear manifold (s), representation by a vector space      60
Linear manifold (s), spanned by given elements      45
Linear manifold (s), zero element      45
Linear, combination of vectors      42
Linear, dependence      48 152-3
Linear, equations      27-30 131-52 180-1
Linear, forms      153
Linear, full linear group      263 268-9
Linear, homogeneous equations      27-30 131 148-52
Linear, operators (operations, transformations)      114-22 277
Linear, substitutions      74 85-87
Linear, transformations of a bilinear form      356
Linear, transformations of a quadratic form      360
Majorization of matrices      328
Mapping      113
Mapping, into      116
Mapping, onto      124
Matrix (matrices), addition      78-79
Matrix (matrices), adjugate      88-90
Matrix (matrices), augmented      140
Matrix (matrices), column type and row type      77
Matrix (matrices), commuting      81
Matrix (matrices), congruent      182
Matrix (matrices), definition      74
Matrix (matrices), determinant of      87
Matrix (matrices), diagonal      76
Matrix (matrices), elementary      170
Matrix (matrices), elementary divisors      194
Matrix (matrices), equations      298-300
Matrix (matrices), equivalence      176
Matrix (matrices), functions      336 342-3
Matrix (matrices), of coefficients      140
Matrix, Gram      156
Matrix, group matrix      273
Matrix, groups      6
Matrix, hermitian      209 301 304
Matrix, idempotent      107
Matrix, index laws      83 93-94
Matrix, invariant properties      122
Matrix, inverse      91 178-9
Matrix, linear manifold of      80
Matrix, multiplication      80-85
Matrix, multiplication by scalars      78
Matrix, normal      305-6 309-10
Matrix, orthogonal      222-9
Matrix, partitioned      100-6
Matrix, periodic      298
Matrix, permutation matrices      271
Matrix, polynomials      97-99 201-8 342-3
Matrix, power series      332-43
Matrix, rank      136-40 158-63
Matrix, rational functions      99-100 201-2 204 5 335
Matrix, representations of groups by matrices      267-72
Matrix, scalar      76
Matrix, sequences      327-9
Matrix, series      330-2
Matrix, similar      119
Matrix, singular and nonsingular      87
Matrix, skew-Hermitian      209 232
Matrix, skew-symmetric      208 227 243-5
Matrix, square      75
Matrix, symmetric      183 209 301-4
Matrix, transposed      76 97
Matrix, triangular      76
Matrix, unit      75
Matrix, unitary      229-32
Matrix, zero      75
Maximal invariant space, of a linear transformation      277-82
Maximal of a group of transformations      282-6
Metric classification of quadrics      385
Minimum polynomial      203-4 207-8 297-8
Minkowski, inequality for positive definite matrices      419-20
Minkowski, theorem on dominated determinants      31-32
Minor, cofactor of      21
Minor, critical      136
Minor, of a determinant      20
Minor, of a matrix      136
Minor, principal      197
Monic polynomial      203
Motion, proper and improper      246
Motion, rigid      246-7
Multiplication table for groups      260-1
Non-commutativity of matrix multiplication      81
Non-negative matrix      328-30
Non-singular matrix      87
Norm of a vector      63
Normal form, reduction of a matrix to      173-6
Normal matrix      305-6 309-10
Normalization of a vector      64
Nullity, Sylvester’s law of      162
Operations, elementary      168-78
Operators, bilinear      353-6
Operators, differential      7 133-4
Operators, linear      114-22 277
Operators, multiplication      172
Operators, polarized      361
Operators, quadratic      357-9 361-2
Oppenheim’s inequality      421
Orthogonal angle      236
orthogonal matrices      222-9
Orthogonal principal vector      237
Orthogonal proper and improper      233
Orthogonal, basis      65-68
Orthogonal, complement      68-69
Orthogonal, congruence group and transformation      266
Orthogonal, group      263
Orthogonal, reduction of quadratic forms      362-3
Orthogonal, reduction of symmetric matrices      4
Orthogonal, set      65-67
Orthogonal, similarity      266
Orthogonal, vectors      64
Orthogonalization process of Schmidt      67-68
Orthonormal bases and sets      65-68 155-6
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