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Turnbull H.W. — An Introduction to the Theory of Canonical Matrices
Turnbull H.W. — An Introduction to the Theory of Canonical Matrices



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Название: An Introduction to the Theory of Canonical Matrices

Автор: Turnbull H.W.

Аннотация:

Thorough and self-contained, this penetrating study of the theory of canonical matrices presents a detailed consideration of all the theory’s principal features. Topics include elementary transformations and bilinear and quadratic forms; canonical reduction of equivalent matrices; subgroups of the group of equivalent transformations; and rational and classical canonical forms. The final chapters explore several methods of canonical reduction, including those of unitary and orthogonal transformations.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1952

Количество страниц: 192

Добавлена в каталог: 19.11.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Absolute invariant      37 154
Adjoint matrix      4 43 161 164
Adjugate determinant      4
Aitken, A. C.      81 142
Algebra, non-commutative      143 150
Alternant      59 60 63 81 158 161
Analytical dynamics      1 161 171 178
Apolarity, vector of      121
Associate matrices      32—33
Associative law      3 20
Augmented matrix      29—30
Autonne      181
Auxiliary unit matrix      62 143—1444 151—155
Axes, principal      1 170
Axisymmetric determinant      99
Axisymmetric determinant, matrix      11 15 32 101 106 164—165
Basis      46 135 142
Basis, change of      117
Bazin      179
Bellavitis      165
Bendixson      110 111
Bibliography of matrices, Muir      178
Bibliography of matrices, Rasch      166
Bicomplementary sets      80
Bilinear forms      15 19
Bilinear forms, differentiated      173
Bilinear forms, families of      30
Bilinear forms, invariant      38
Bilinear forms, pencil of      114 129—131
Bilinear forms, reciprocal      162
Bilinear forms, singular pencil of      129—130
Bilinear forms, symmetric      83
Binary p-ic matrix      21
Binary p-ic matrix, quantic      56
Binary p-ic matrix, vectors      119
Binet      31 73
Binet — Cauchy theorem      9 25 28
Bordered determinant      161
Born      178
Brauer      110 111
Brioschi      98 111
Bromwich      110 111 142
Brown, O. E.      142
Browne, E. T.      110 111 183 184
Burnside      56
Canonical reductions of matrices, collineatory, classical      58 60 75
Canonical reductions of matrices, collineatory, of Jacobi      64 80 81
Canonical reductions of matrices, collineatory, rational      45 49
Canonical reductions of matrices, collineatory, real      72
Canonical reductions of matrices, congruent, of Jacobi      94
Canonical reductions of matrices, congruent, rational      138—139
Canonical reductions of matrices, equivalent, of Smith      22—23
Canonical reductions of matrices, geometrical meaning of      53—54
Canonical reductions of matrices, uniqueness of classical      67
Canonical reductions of pencil of quadratics      115—116 132—135 141 142 182
Canonical reductions of pencils, classical      116
Canonical reductions of pencils, rational      116
Canonical reductions of pencils, real      72 182
Canonical reductions of singular pencils      125—135
Cauchy      9 25 26 31 73 111
Cayley      9 30 56 107 112 166 178
Cayley matrix operator      178
Cayley matrix operator, orthogonant      107
Cayley — Hamilton theorem      43—44 180
Chains of isolated elements      60 69
Chains of isolated elements, of vectors      49
Characteristic determinant      41
Characteristic determinant, equation      41
Characteristic determinant, function      41
Characteristic determinant, function of vector      43
Characteristic determinant, reduced, equation      48
Characteristic determinant, reduced, function      46—52 161—162
Characteristic determinant, Segre      61 64 70 79
Christoffel      111
Chrystal      178
Cipolla      99
Classical canonical form      58 60 75
Clebsch      111
Cogredient variables      37 38
Collineation      36
Collineation, as successive polar reciprocation      153 182
Collineation, geometrical meaning      40 42
Collineation, reciprocal      40
Collineation, singular      40 142
Collineatory sub-group      35 37
Collineatory sub-group, transformation      11 12 13
Combinants      113
Commutantat functions      148
Complete diagonal      65
Compound commutants      148
Compound commutants, determinants      28
Compound commutants, matrices      27 73 90 93 98 180
Compound commutants, quadratic forms      91
Conditioned maxima and minima      170
Confluent alternant      60 63 81
Confluent alternant, difference-product      60 63 75 81 150
Conformable matrices      3
Congruent      See Canonical reductions
Congruent, group      36—37 82
Congruent, reduction of Hermitian form      84
Congruent, transformation      11 12 13
Conics      1 82
Conics, congruent reduction      82
Conics, pencils of      115 118 182
Conjugate matrices      32—33
Conjugate matrices, partitions      80
Conjunctive reduction of skew form      87
Conjunctive reduction of skew form, sub-group      36
Continued fractions      74 75 182
Contragredient variables      37
Convergence of matrix functions      73 81
Correlation      37
Correlation, canonical form of      84 138—139 140—141
Correlation, statistical      174—175 178
Courant      178
Cpmmutants      143 145 147—149
Criteria for denniteness      91
Criteria for denniteness, for rank      92—93
Darboux      142 164 165 166
Definiteness      89—90
Definiteness, criteria for      91
Delta, Kronecker      151
Dependence of vectors      45
Dependence of vectors, of vector pencils      119
Determinant      3 4 28 41
Determinant, alternant      51 60 63 75 81
Determinant, characteristic      41
Determinant, compound      28
Determinant, recurrent      56
Determinant, skew symmetric      99
Determinant, with matrix elements      76
Diagonal, complete      65
Diagonal, secondary      11
Diagonatthatrix      7—8
Diagonatthatrix, partitioning      7
Dickson      56 81 84 141 142
Difference equations      176—177
Difference with matrix factors      77—78
Difference-product      59
Difference-product, confluent      60 63 75 81
Differential equations      1 176—177
Differentiated bilinear form      173
Differentiated bilinear form, matrix      180
Dirac      151 166
Dual sets of integers      80
Dynamical equilibrium      161 171 178
Elementartheiler      30 69 108 180
Elementary divisors      69 108 180
Elementary divisors, matrix      48 56—57 62
Elementary divisors, of pencil      117 136 180
Elementary divisors, transformations      10—13
Elements, isolated      8
Ellipse, semi-axes of      170 171
Ellipsoid      171
Equations, difference      176—177
Equations, differential      176—177
Equations, incompatible      29
Equations, linear algebraic      10 13 29
Equations, linear matrix      143 145 148 166
Equations, normal, of Least Squares      173
Equations, singular linear algebraic      29
Equilibrium, dynamical      161 171 178
Equivalent matrices      19
Equivalent matrices, $\lambda$-matrices      26
Equivalent matrices, pencils      116—117
Equivalent matrices, singular pencils      129
Euclid      18 22
Factors, invariant      27 30
Factors, matrix and scalar      15
Factors, of pencils      116—117
Factors, of pencils of Hermitian forms      119
Factors, symmetric, of general matrix      152—153
Families of forms      30 See
Ferrers      80
Fibonacci numbers      75
Field      20 45
Fischer      99
Forms, bilinear and quadratic      15
Forms, families of      30
Forms, hermitian      34
Forms, invariant or latent      37—39
Forms, linear      45
Forms, pairs of      106—107
Forms, pencils of      114 130—131
Franklin      183
Function, generating, of key quadratics      158—159
Function, maxima and minima of real      168—169
Function, of matrix      7 73 81 149
Function, of normal frequency      174—175 See
Functional equation of resohent      161
Gauss      18 98
Generalized rotation      102
Geometrical meaning of collineation      40 42 53 54 153
Geometrical meaning of collineation, of congruent group      82 115 181 182
Geometry, co-ordinate      1 9 170—171
Grade of vector      47 53
Group      20
Group, properties      20 34—35
Group, sub-groups of equivalent      19 32—36
H.C.F. process      17
H.C.F. process for polynomials      16 21—22
Hadamard's Theorem      97 98
Hamilton      9 43 44
Hawkes      81
Heisenberg      178
Hensel      81
hermite      18 44 111
Hermite polynomials      102
Hermitian matrices and forms      32—34 44
Hermitian matrices and forms, congruent reduction of      84
Hermitian matrices and forms, invariant factors of      119
Hermitian matrices and forms, latent roots of      101 106
Hermitian matrices and forms, pencils      114 130—131 166
Hermitian matrices and forms, reduction of      133—134 141 142
Hermitian matrices and forms, singular case      134—135
Hermitian matrices and forms, skew      33—34 98
hessian      18
hilbert      161 178
Hilton      142 166
Hirsch      110 111
Hitchcock      166
Image points      40
Imaginary matrices      33 34
In variance of minimal indices      120—121
Ince      178
Incompatible equations      29
Indicatrix      167
Indices, minimal      120—121
Inequalities in principal minors of definite determinant      98 177
Inertia, law of      89
Infinite matrices      3 166
Ingraham, M. H.      142
Inner product      3 15 37 38
Interpolation formula of Lagrange      76
Interpolation formula of Lagrange, of Sylvester for matrices      76—77 81 160
Invariant bilinear forms      38 143
Invariant bilinear forms, quadratic forms      154
Invariant factors      27 30
Invariant factors, of pencil      116 117 119
Invariants      37 154
Invariants, of singular pencil      128
Inverse factorial series      74
Involutory matrices      33 103
Isolated elements and minors      8
Isolated elements and minors, semi-isolated      8
Isotropic vector      103
jacobi      44 64 80 81 94 98 99
Jacobi, collineatory canonical form of      64
Jacobi, congruent canonical form of      94
Jessop      166
Jordan, C.      80 181
Jordan, P.      178
Kantor, S.      142
Kelvin      99
kinetic energy      171
Kowalewski      56
kronecker      98 141 142
Kronecker, delta      151
Lagrange      98 170 172
Lagrange, $\lambda$-matrices      14 16 21 44 54 55 176
Lagrange, interpolation formula      76
Lagrange, undetermined multipliers      170
Landsberg      56 81
Latent forms      37 38 39 143 154
Latent forms, matrices      106 159
Latent forms, points      44
Latent forms, primes      154
Latent forms, quadratic forms      155
Latent roots, theorems on, definition and general      41 44
Latent roots, theorems on, inequalities in      110 111 183 184
Latent roots, theorems on, of Hermitian and real symmetric      101 106
Latent roots, theorems on, of Hermitian principal minors      101 172
Latent roots, theorems on, of induced and compound matrices      73 183
Latent roots, theorems on, of orthogonal and unitary      107 108
Latent roots, theorems on, of orthogonal principal minors      110 111
Latent roots, theorems on, of skew and skew Hermitian      101 106
Lattes      56
Law, associative      3 20
Law, commutative      3
Law, of inertia      89
Law, reversal      4
Least squares      9 173
Ledermann, W.      142
Legendre polynomials      102
Linear algebraic equations      10 13 29
Linear algebraic equations, dependence      45
Linear algebraic equations, difference equations      176—177
Linear algebraic equations, differential equations      176—177
Linear algebraic equations, forms      45
Linear algebraic equations, matrix equations      143 145 148 166
Linear algebraic equations, transformations      13
Loewy      110 111
Logsdon      141
MacDuffee, C. C.      179
Matrices, combinantal      113
Matrices, conformable      3
Matrices, pencils of      113
Matrices, permutable      116 180
Matrix, adjoint      4 161 164
Matrix, alternant      59 158 161
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