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| Cullis C.E. — Matrices and Determinoids |
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| Предметный указатель |
Equality of matrices, conventional (dissimilar matrices) 154
Equality of matrices, Conventionally equal matrices 154—6 161 173 204
Equality of matrices, Conventionally equal matrices are equivalent 156 173 204
Equality of matrices, identical (similar matrices) 154
Equations, Cancellation of factors in 357 325 336 354
Equations, Connections between 364 412 415
Equations, Equations with only extravagant solutions 391 409—10
Equations, Extravagant equations 388—92
Equations, Linear algebraic equations 364—415
Equations, Manipulation of 156—7 176—7 357
Equations, Matrix equations 156 176 299—358
Equations, Matrix equations of the first degree, AX=0 324—6
Equations, Matrix equations of the first degree, AX=C 302—5 312—27
Equations, Matrix equations of the first degree, AXB=0 354—6 (see also “Matrix equations of the first degree”)
Equations, Matrix equations of the first degree, AXB=C 309—11 337—57
Equations, Matrix equations of the first degree, X = A 300
Equations, Matrix equations of the first degree, X+A=B 302
Equations, Matrix equations of the first degree, XB=0 335—7
Equations, Matrix equations of the first degree, XB=C 306—9 327—37
Equipotent 297
Equipotent matrices 285 297
Equipotent matrices, have equal ranks 283 286 297
Equivalence of conventionally equal matrices 173
Expansion of bordered detant 240
Expansion of bordered detant of bordered detoid 241
Expansions of a detoid in terms of elements of a long row 105
Expansions of a detoid in terms of elements of a long row, generalisation 110
Expansions of a detoid in terms of simple minor detants 113
Expansions of a detoid in terms of simple minor detants of a long-cut minor matrix 116
Expansions of a detoid in terms of simple minor detants of a long-cut minor matrix, generalisations 121—3 143
Expansions of a detoid in terms of simple minor detants, generalisation 123
Expressions for, affects of derived sequences 57
Expressions for, detoid of a product of two matrices 210 215—6 220
Expressions for, detoid of any matrix product 228 233 243
Expressions for, elements of a product matrix 159 162—3 194
Extended derived matrix 90—1
Extended derived matrix, derived product 22 29
Extended derived matrix, derived sequence 56
Extended derived matrix, derived sequence, Affects unaltered by extension 91 29 56
Extravagant equations 388—92
Extravagant matrix, defined 288
Extravagant solutions of linear equations 404
Extravagant solutions of linear equations are self-orthogonal 405
Factor matrices, defined 158 190
Factors of a derived product 3
Factors of a derived product, relative affects of 24
Factors of a derived product, Sign of product independent of the order of its factors 13 47
Factors of special form in a matrix product, quasi-scalar matrix 179
Factors of special form in a matrix product, scalar matrix 178 207 219
Factors of special form in a matrix product, scalar number 157 206
Factors of special form in a matrix product, square matrix whose order is equal to the efficiency 224
Factors of special form in a matrix product, unit matrix 177 207 219
Factors of special form in a matrix product, zero matrix 207
Factors, cancellation of in a matrix equation 357 325 336 354
Factors, cancellation of in a matrix identity 358
Formation (see “Schemes of formation”)
Forward moves, applied 49 50 62 91
Forward moves, defined 49 57
Forward steps, applied 12 24 55 56
Forward steps, defined 4 55
Functional dependences 415
Functional matrices, Rank and connections of 295
Fundamental matrix or detoid 86 248
Fundamental matrix or detoid, sequence 55
General solution of a matrix equation, formulae for AX=0 324
General solution of a matrix equation, formulae for AX=C 317 319 303
General solution of a matrix equation, formulae for AXB=0 354
General solution of a matrix equation, formulae for AXB=C 345 349 310
General solution of a matrix equation, formulae for XB=0 336
General solution of a matrix equation, formulae for XB=C 330 331 306
General solution of a system of linear algebraic equations, formulae for any system 368—9 375—6 400-3
General solution of a system of linear algebraic equations, formulae for homogeneous system 381—2 392
Generalisation of a determinoid 48 134 136—7 244
Generalisation of Laplace’s development of a determinant 119 123 143 Sum.”)
Homogeneous linear equations (see “System”)
Horizontal suffixes 12
Horizontal suffixes, affects 24
Horizontal suffixes, minors 233—4 249
Horizontal suffixes, moves 48
Horizontal suffixes, steps 4
Identical equality of matrices 154
Identical long rows in a detoid 20 44
Identity, Cancellation of matrix factors in a matrix identity 358
Incomplete derived product 4 22
Incomplete derived product, affect not altered 29
Incomplete derived product, extended or completed 22 23
Incomplete derived sequence 55
Incomplete derived sequence, affect not altered 56
Incomplete derived sequence, extended or completed 56
Inferior simple minors 115 134 136—7 143
Inner elements of a sequence 76
Inner elements of a sequence, rows and elements of a matrix 39 44
Interchange of two elements in a derived product 13 27 47
Interchange of two elements in a sequence 72 76
Interchange of two elements in a sequence, inner and outer elements 76
Interchange of two long rows changes the sign of the fundamental detoid 20 36 43
Interchange of two long rows changes the sign of very complete derived product 44
Interchange of two long rows changes the sign of very simple minor detant 44
Interchange of two long rows changes the sign of very superior short-cut minor detoid 44
Interchange of two long rows in a detoid 20 36 43
Interchange of two parallel rows in a matrix 30 38
Interchange of two parallel rows in a matrix, inner and outer elements and rows 39
Interchange of two short rows in a detoid 44
Interchange of two suffixes in a matrix 36 44
Interchange of two suffixes in a matrix, inner and outer elements and rows 44
Invariance of the sign of a derived product 47
Invariance of the sign of a derived product of a complete derived product 13
Inverse matrices, defined 183
Inverse matrices, inverse of non-singular matrix 267
Inverse matrices, inverse post-factors 183 322
Inverse matrices, inverse pre-factors 183 335
Inverse matrices, principal inverse matrix 182
Inverse matrices, unique inverse of undegenerate square matrix 183 267 309
Inversions in a derived sequence 60
Inversions in a derived sequence, Affect determined by counting them 60
Irreducible matrix equations, defined 312 328 338
Irreducible matrix equations, Solutions when finitely solvable 323 333 352
Laplace’s development of a determinant, generalisations of 119 123 143 Case
Law of equality for matrices 153
Law of equality for matrices, Associative law for addition of matrices 155—6
Law of equality for matrices, Associative law for addition of matrices for matrix products 184 186 188
Law of equality for matrices, Commutative law for addition of matrices 155—6
Law of equality for matrices, Commutative law for addition of matrices not true for multiplication 174 208
Law of equality for matrices, Distributive law for multiplication by a scalar number 157
Law of equality for matrices, Distributive law for multiplication by a scalar number for matrix products 173 184 188—9
Leading element, line, row, diagonal, position 1 2 16 55
Leading element, line, row, diagonal, position, Moves which bring to leading position, a derived matrix 91—4
Leading element, line, row, diagonal, position, Moves which bring to leading position, a derived product 49
Leading element, line, row, diagonal, position, Moves which bring to leading position, a derived sequence 62—4
Linear algebraic equations 364—415
Linear functions, connections between 364
Long rows, Addition of detoids 107
Long rows, Addition to one long row of scalar multiples of other long rows 109
Long rows, defined 1 16
Long rows, Expansion of a detoid in terms of elements of a long row 105
Long rows, Expansion of a detoid in terms of the simple minor detants of a long-cut minor 119
Long rows, Interchange of two long rows 20 36 43
Long rows, Long-cut minors 115—23 136-8
Long rows, Manipulation of matrix equations 156—7 176—7 357
Long rows, Multiplication of a detoid by a scalar number 19 108
Long rows, Properties of 19—21 36 43—4 105-9
Long rows, Property of any two long rows 108 110
Long rows, Property of any two similar long-cut minors 122 123
Long rows, Sum of long-cut simple minor detoids of given reduced order 136
Long rows, Two long rows identical 20 44
Long rows, Vanishing of a long row 107
Manipulation of 156—7 176—7 357
Matrices of minor detants 248 253
Matrices of minor detants, standard 251 (see “Complete matrices of minor detants”)
Matrices, addition and subtraction of 155—6
| Matrices, augmented and unaugmented 8 287 312 327 337 366 374
Matrices, degenerate and undegenerate 265
Matrices, equality of 153—5
Matrices, inverse 182—3
Matrices, quipotent 288 297
Matrices, similar 153—5
Matrices, singular and non-singular 266
Matrix equations 156 176 299—358
Matrix equations of the first degree 299—358
Matrix equations of the first degree, defined 299 300
Matrix equations of the first degree, The equation X+A—B 302
Matrix equations of the first degree, The equation X=A 300
Matrix equations of the first degree, The equations AX=0, XB=0, AXB=0, Conditions for non-zero solutions 325 336 354
Matrix equations of the first degree, The equations AX=0, XB=0, AXB=0, General solutions 304 324 307 335 310 354
Matrix equations of the first degree, The equations AX=0, XB=0, AXB=0, Ranks of solutions 325 336 355—6
Matrix equations of the first degree, The equations AX=C, XB=C, AXB=C, Augmented and unaugmented matrices 312 327 337
Matrix equations of the first degree, The equations AX=C, XB=C, AXB=C, Conditions for finite solvability 314 329 341
Matrix equations of the first degree, The equations AX=C, XB=C, AXB=C, Formulae for general solutions 317 319 330 331 345 349
Matrix equations of the first degree, The equations AX=C, XB=C, AXB=C, Infinite solutions 326 337 356
Matrix equations of the first degree, The equations AX=C, XB=C, AXB=C, Reduction to irreducible equations 312 328 338
Matrix equations of the first degree, The equations AX=C, XB=C, AXB=C, Simplest particular solutions 317 319 330 332 345 349
Matrix equations of the first degree, The equations AX=C, XB=C, AXB=C, Solutions when A and B are undegenerate square matrices 302 306 309
Matrix equations of the first degree, The equations AX=C, XB=C, AXB=C, Solutions when irreducible and finitely solvable 323 333 352
Matrix equations, Cancellation of factors in 357
Matrix factors, Cancellation of 357—63
Matrix factors, defined 158 190
Matrix factors, Special factors, quasi-scalar matrix 179
Matrix factors, Special factors, scalar matrix 178 207 219
Matrix factors, Special factors, square matrix whose order is equal to the efficiency of the product 224
Matrix factors, Special factors, unit matrix 177 207 219
Matrix factors, Special factors, zero matrix 207
Matrix identities, cancellation of factors in 358—63
Matrix products are associative 186 188
Matrix products are distributive 173 188—9
Matrix products are in general not commutative 174 208
Matrix products, Conjugates of 175 205
Matrix products, defined 158—64 184—90
Matrix products, Distribution over partial products 164 167 195 199
Matrix products, Properties of active rows 168—72 199—204
Matrix products, Properties of passive rows 164—8 195-9
Matrix products, Reciprocals of standard products of square matrices 261
Matrix products, Standard forms of 158 184 192
Matrix, cannot be equated to a non-zero scalar number 179
Matrix, conjugate 5
Matrix, conjugate reciprocal 111 180
Matrix, defined 1 153—64 184—90
Matrix, functional 295
Matrix, inverse, principal inverse 182—3
Matrix, quasi-scalar 179
Matrix, reciprocal 110—11
Matrix, rectangular 1
Matrix, scalar 178
Matrix, self-conjugate or symmetrical 5
Matrix, square 1
Matrix, undegenerate square 183 267 289 309
Matrix, unit 11 177
Matrix, zero 265 (see also “Rectangular matrix” “Square
Minor determinants, Affected minor detant is invariant for all derangements of its rows 101
Minor determinants, Affects of, theorems concerning 96—8 100-1
Minor determinants, Co-factors of 119
Minor determinants, Complete matrices of 248 253
Minor determinants, Complete matrices of distinct 115 249
Minor determinants, Complete matrices of of a long-cut minor matrix 122—3
Minor determinants, Complete matrices of of a short-cut minor matrix 138
Minor determinants, Sum of affected simple minor detants 115 209
Minor determinants, Sum of affected simple minor detants, affected minor detants of order k 134 136—7 244
Minor matrices and determinoids (cont.), Sum of products of a minor detoid and its co-factor belonging to two fixed complementary simple minors 140—3 107 116 138
Minor matrices and determinoids, Affected minor detoid invariant for derangements of its long rows 101
Minor matrices and determinoids, Affected minor detoid invariant for derangements of its long rows, affects unaltered by extension 91
Minor matrices and determinoids, Affected minor detoid invariant for derangements of its long rows, brought to leading position by moves 91—4
Minor matrices and determinoids, Affected minor detoid invariant for derangements of its long rows, complementary 102—4
Minor matrices and determinoids, Affected minor detoid invariant for derangements of its long rows, corranged and deranged 2 16
Minor matrices and determinoids, Affected minor detoid invariant for derangements of its long rows, extended and completed 90 91
Minor matrices and determinoids, Affects of, defined 86—90
Minor matrices and determinoids, Co-factors of minor detoids 141 106 119
Minor matrices and determinoids, Co-factors of minor detoids of simple minor detoids 145
Minor matrices and determinoids, defined 2 16
Minor matrices and determinoids, Simple minors 3 115 116
Minor matrices and determinoids, Simple minors, horizontal and vertical 233 234 249
Minor matrices and determinoids, Simple minors, inferior 116 134 136
Minor matrices and determinoids, Simple minors, long-cut 116 123 136
Minor matrices and determinoids, Simple minors, reduced and unreduced orders of 116
Minor matrices and determinoids, Simple minors, short-cut 116 123 134 138
Minor matrices and determinoids, Simple minors, superior 116 123
Minor matrices and determinoids, Theorems concerning 90—104
Minor sequences, Affects of, defined 56
Minor sequences, Affects of, defined of complementary minors 70
Minor sequences, Affects of, defined, Theorems concerning 62—71
Minor sequences, affects unaltered by extension 56
Minor sequences, brought to leading position by moves 62—4
Minor sequences, complementary 56 70 71
Minor sequences, complete 59
Minor sequences, complete and incomplete 55
Minor sequences, corranged and deranged 55
Minor sequences, defined 55
Minor sequences, distinct 61
Minor sequences, Expressions for affects 57—62
Minor sequences, Expressions for affects of corranged minors 59 61
Minor sequences, Expressions for affects of corranged minors of a sequence of natural numbers 61
Minor sequences, extended and completed 56
Minor sequences, Inversions in 60 (see “Affects of derived sequences”)
Minor sequences, standard arrangement of 251
Moves in a matrix, defined 48
Moves in a matrix, forward and backward 49
Moves in a matrix, horizontal and vertical 48
Moves in a matrix, moves bringing to leading position, a derived matrix 91—3
Moves in a matrix, moves bringing to leading position, a derived matrix, affect thereby determined 91
Moves in a matrix, moves bringing to leading position, a derived product 49
Moves in a matrix, moves bringing to leading position, a derived product, affect thereby determined 49
Moves in a sequence, defined 57
Moves in a sequence, forward and backward 57
Moves in a sequence, its affect thereby determined 49
Moves in a sequence, minor sequence brought to leading position by moves 49
Multiplication by or into it does not alter rank 283
Multiplication of 158—64 184—90
Multiplication of a matrix by a scalar number 157
Multiplication of a matrix by a scalar number is commutative and distributive 157
Multiplication of a matrix by a scalar number of a determinoid by a scalar number 19 108
Multiplication of a matrix by a scalar number of a matrix product by a scalar number 206
Multiplication of matrices 158—64 184—90
Multiplication of matrices is associative and distributive but in general not commutative 173—4 184—90 208
Mutually orthogonal solutions of a system of homogeneous equations 404
Mutually orthogonal solutions of any system of linear equations 411
Natural numbers, sequences of 60
Natural numbers, sequences of, affect of a corranged minor 61
Natural numbers, sequences of, affect of any minor 60
Non-commutative property of matrix products 174 208
Non-extravagant equations 388—92 399
Non-extravagant mutually orthogonal solutions of a system of homogeneous linear equations 405—7
Non-singular matrix, Beciprocal, conjugate reciprocal and inverse of 267
Non-singular matrix, defined 266
Non-zero solutions, conditions for 325 336 354 381
Notation for a scalar matrix 178
Notation for a unit matrix 11 177
Notation for augmented matrices 8
Notation for detoids 15 17 347
Notation for matrices 1 5 347
Notation for sequence 55 57
Order of a square matrix and detant 1 16
Orders of a matrix, horizontal and vertical 1
Orders of a matrix, horizontal and vertical of a detoid 1 16
Orders of a matrix, horizontal and vertical, Reduced and unreduced orders of a simple minor 116
Orthogonal solutions of a system of linear algebraic equations, complete sets of 406 407 411
Orthogonal solutions of a system of linear algebraic equations, Condition that all solutions shall be mutually orthogonal 409
Orthogonal solutions of a system of linear algebraic equations, defined 405 412
Outer elements of a sequence 76
Outer elements of a sequence, elements and rows of a matrix 39
Parallel rows, Interchanges of 30—44 (see “Interchange”)
Partial products, Matrix product expressed as a sum of 164 167 195 199
Partial products, product expressed as sum of 164 167
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