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| Авторизация |
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| Cullis C.E. — Matrices and Determinoids |
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| Предметный указатель |
 , defined 127
, Properties of 127—34
, Table of values 131
 127
Abbreviated notations for affects 66
Abbreviated notations for detoids 17—19 347
Abbreviated notations for matrices 1 5—12 347
Abbreviated notations for matrices, a scalar matrix 178
Abbreviated notations for matrices, a unit matrix 11 177
Abbreviated notations for matrices, augmented matrices 8
Active and passive rows, defined in a product of two matrices 158
Active and passive rows, defined in any matrix product 190
Active and passive rows, defined, Corresponding rows 160 163 191
Active rows, properties of, in a product of two matrices 168
Active rows, properties of, in any matrix product 199 222
Activities of a matrix product 158 191
Addition and subtraction of matrices is commutative and associative 155—6
Addition and subtraction of matrices, dissimilar matrices 156
Addition and subtraction of matrices, Equivalence of conventionally equal matrices 156
Addition and subtraction of matrices, similar matrices 155
Addition of detoids 107
Affected derived products 24
Affected derived products, Sums of 27 134 136
Affected elements of a matrix vi.
Affected minor detants, Complete matrices of 252—3 258
Affected minor detants, Sums of 115 134—7 209 221 244
Affected minor detants, Sums of products of 210 217 225 228 244
Affected simple minor detoids, Sums of 123 134—7
Affects of derived or minor matrices of complementary minors 102
Affects of derived or minor matrices, compared with affect of leading product 89 91
Affects of derived or minor matrices, defined by affects of sequences 86
Affects of derived or minor matrices, obtained by counting moves 91
Affects of derived or minor matrices, Theorems concerning 91—104
Affects of derived or minor matrices, unaltered by extension or completion 91
Affects of derived or minor sequences of complementary minor sequences 70 71
Affects of derived or minor sequences of complete derived sequences 59
Affects of derived or minor sequences of corranged minor sequences 58 59 61
Affects of derived or minor sequences, Changes in, due to an interchange of two elements 72 76
Affects of derived or minor sequences, defined by counting steps 56
Affects of derived or minor sequences, Expressions for 57—61
Affects of derived or minor sequences, obtained by counting inversions 60
Affects of derived or minor sequences, obtained by counting moves 62
Affects of derived or minor sequences, obtained by reducing the fundamental sequence to a sequence of natural numbers 60
Affects of derived or minor sequences, relative affects 56
Affects of derived or minor sequences, Theorems concerning 62—71
Affects of derived or minor sequences, unaltered by extension or completion 56
Affects of derived products (cont.), Affects of complete derived products 27
Affects of derived products (cont.), Changes in, due to interchange of two parallel rows 30 38
Affects of derived products (cont.), Changes in, due to interchange of two parallel rows to interchange of two suffixes 36 44
Affects of derived products (cont.), Changes in, due to interchange of two parallel rows to reversals of rows 50
Affects of derived products (cont.), obtained by counting moves 50
Affects of derived products (cont.), reduced to affects of sequences 83
Affects of derived products (cont.), Rules for determining 25—30
Affects of derived products (cont.), Signs determined by affects 24
Affects of derived products (cont.), Signs determined by affects, independent of order of factors 47
Affects of derived products (cont.), Theorems concerning 96—8 100
Affects of derived products (cont.), unaltered by extension or completion 29
Affects of derived products, defined by counting steps 13 24
Affects of derived products, defined by counting steps, relative affects 24
Affects, horizontal, vertical and total vi 24 56 86
Affects, horizontal, vertical and total, Relative affects 24 56
Affects, horizontal, vertical and total, Signs determined by vi 24
Algebraic equations (see “Systems of linear algebraic equations”)
Algebraical sum (see “Sum”)
Associative law for addition and subtraction of matrices 155—6
Associative law for matrix products 184 186 188
Augmented matrices of matrix equations 312 327 337
Augmented matrices of systems of linear algebraic equations 366 374
Augmented matrices, abbreviated notations for 8
Augmented matrices, Banks of 287
Augmented matrices, Generalisation of 347
Augmented solutions of linear equations 395 399 411
Backward steps 4 16 55
Backward steps, moves 49
Bordered detants 240
Bordered detants, detoids 241
Cancellation of a matrix factor in a matrix equation 357 325 336 354
Cancellation of a matrix factor in a matrix identity 358
Cancellation of a matrix factor of a scalar factor in a matrix equation 157
Cancellation of, in matrix equations 357—8 336 354
Cancellation of, in matrix identities 358—63
Chain of matrix factors 190—3
Changes in the affect and sign of a derived product caused by an interchange of two rows or two suffixes 30 36 38 44
Changes in the affects of derived sequences caused by an interchange of two elements or two suffixes 72 76
Classification of simple minors 115
Co-factor of an element of a matrix 106
Co-factor of an element of a matrix of a minor detant 119
Co-factor of an element of a matrix of a minor detoid 141
Co-factor of an element of a matrix of a simple minor detoid 145
Co-factor of an element of a matrix, Sum of products of an element and its co-factor 107 140
Co-factor of an element of a matrix, Sum of products of an element and its co-factor of a minor detant and its co-factor 119 138
Co-factor of an element of a matrix, Sum of products of an element and its co-factor of a minor detoid and its co-factor 143
Co-factor of an element of a matrix, Sum of products of an element and its co-factor, Generalisations 108 123
Commutative law for addition of matrices 155—6
Commutative law, not true for matrix products 174 208
Commutative products 175 181—3 262-3
Commutative products, product of a square matrix and its conjugate reciprocal 181 263
Commutative products, product of an undegenerate square matrix and its inverse 183
Commutative products, product of two square matrices when the product is a non-zero scalar matrix 262—3 266 309
Complement of a simple minor detoid 145
Complementary corranged minor detants of a square matrix have like signs 104
Complementary minor matrices or detoids, Affects of 102 104
Complementary minor matrices or detoids, Complementary corranged minor detants of a square matrix have like signs 104
Complementary minor matrices or detoids, defined 102
Complementary minor matrices or detoids, Sums of affected products of complementary minor detoids 140 116 138
Complementary minor sequences, Affects of 70 71
Complementary minor sequences, defined 56
Complete derived products, affects and signs of 27
Complete derived products, defined 3 22
Complete derived products, signs independent of order of factors 13 27 47
Complete derived products, sum of 12 27
Complete derived sequences, affects of 59 62—3
Complete derived sequences, defined 55
Complete matrices of the minor detants of a given fundamental matrix 248
Complete matrices of the minor detants of a given fundamental matrix of affected minor detants 252
Complete matrices of the minor detants of a given fundamental matrix of unaffected minor detants 249
Complete matrices of the minor detants of a given fundamental matrix, Banks of 289
Complete matrices of the minor detants of a given fundamental matrix, Schemes of formation for 249 252
Complete matrices of the minor detants of a given fundamental matrix, Schemes of formation for Common schemes 250
Complete matrices of the minor detants of a given fundamental matrix, Schemes of formation for Standard schemes 251
Complete matrices of the minor detants of a given fundamental matrix, Standard complete matrix of minor detants 251
Complete matrices of the minor detants of a given matrix product 253
Complete matrices of the minor detants of a given matrix product, Corresponding schemes of formation for 256
Complete matrices of the minor detants of a given matrix product, Standard equations for 258
Complete sets of unconnected solutions of a system of homogeneous linear equations 386
Complete sets of unconnected solutions of a system of homogeneous linear equations, Complete sets of mutually orthogonal solutions 406—7
Complete sets of unconnected solutions of a system of homogeneous linear equations, Determination of all possible 388
Complete sets of unconnected solutions of any system of linear equations 395 397
Complete sets of unconnected solutions of any system of linear equations, Complete sets of mutually orthogonal solutions 411
Complete sets of unconnected solutions of any system of linear equations, Determination of all possible 399
Completed derived products 22 29
Completed derived products, Affects not altered by completion 29
Completed minor matrices and detoids 90 91
Completed minor matrices and detoids, Affects not altered by completion 91
Completed minor sequences 56
Completed minor sequences, Affects not altered by completion 56
Conditions for non-zero solutions of a homogeneous matrix equation 325 336 354
Conditions for solvability of a matrix equation 301—2 314 327 329 337 341 357
Conditions for solvability of a system of linear algebraic equations 366 374 380—1
Conditions for solvability, for finite solutions, for infinite solutions 366 374
Conditions that a matrix may have rank r 288
Conjugate matrix or detoid, A detoid and its conjugate are equal 16
Conjugate matrix or detoid, defined 5
Conjugate matrix or detoid, Rank of 266
Conjugate matrix or detoid, Rank of a product of two mutually conjugate matrices 288
Conjugate matrix or detoid, Self-conjugate or symmetrical 5
Conjugate of a matrix product 175 205
Conjugate reciprocal matrix or detoid 111 180
Conjugate reciprocal matrix or detoid, Product of a matrix and its conjugate reciprocal 180 (see also “Beciprocal matrix”)
Conjugate reciprocal of a square matrix 181 267 309
| Conjugate reciprocal of a square matrix, Product of a square matrix and its conjugate reciprocal 181 263
Conjugate reciprocal of a square matrix, Rank of 289
Conjugate reciprocal of a standard product of square matrices 261
Connections between 364 412 415
Connections between linear equations 364 412 415
Connections between linear equations, between linear functions 364
Connections between the rows of a matrix 269
Connections between the rows of a matrix of a functional matrix 294
Connections between the rows of a matrix of a functional matrix, Theorems concerning 271—8 287
Connections between the rows of a matrix of a functional matrix, Unconnected connections 326 337
Conventional equality of matrices 154
Conventionally equal matrices 154—7 161 173 204
Conventionally equal matrices are equivalent 156 173 204
Corranged and deranged minors 2 55
Corranged minors of a matrix 2
Corranged minors of a matrix, Affects of complementary 102 104
Corranged minors of a sequence 55
Corranged minors of a sequence, Affects of complementary 70 71
Corranged minors of a sequence, Affects of corranged minors of a sequence of natural numbers 61
Corresponding rows in a matrix product 160 163 191
Corresponding schemes of formation of matrices of minor detants 256
Criteria for the rank of a matrix 279 280 286 288
Criteria for the rank of a matrix has no inverse 267
Criteria for the rank of a matrix, Degenerate matrix: defined 265
Definitions, augmented and unaugmented matrices 366 374
Definitions, augmented and unaugmented solutions 395
Definitions, finite solutions 365
Definitions, infinite solutions 403
Definitions, orthogonal solutions 412
Degenerate matrix has no inverse 3 22 335
Degenerate matrix has no inverse, Product of undegenerate square matrix and its inverse is commutative 183
Dependences between functions 415
Derived determinants, Theorems concerning affects of 96—8 100-1
Derived matrices and determinoids, Affects of 86
Derived matrices and determinoids, Affects of Affects unaltered by extension or completion 91
Derived matrices and determinoids, Affects of Theorems concerning affects 91—104
Derived matrices and determinoids, brought to leading position by moves 91—3
Derived matrices and determinoids, complementary 102 104
Derived matrices and determinoids, corranged and deranged 2
Derived matrices and determinoids, defined 2 16
Derived matrices and determinoids, extended and completed 90 91
Derived products independent of order of factors 47
Derived products, affected and unaffected 24
Derived products, Affects not altered by extensions 29
Derived products, Affects of, relative affects 24
Derived products, complete 12 13
Derived products, complete and incomplete 3 4 22 27 47
Derived products, defined 3 22
Derived products, extended and completed 22 29
Derived products, reduced to leading product by moves 48
Derived products, Signs determined by affects 24
Derived products, Sum of affected complete products 12 27
Derived products, Sum of affected products of order k 48 134 136—7
Derived products, Theorems concerning affects 96—8 100
Derived sequences, Affects of 56
Derived sequences, brought to leading position by moves 62—4
Derived sequences, complementary 56 70 71
Derived sequences, complete 59
Derived sequences, complete and incomplete 55
Derived sequences, corranged 58—9 61 65 70-1
Derived sequences, corranged and deranged 56
Derived sequences, defined 55
Derived sequences, extended and completed 56
Derived sequences, Inversions in 60
Derived sequences, not altered by extensions 56
Derived sequences, Signs determined by affects of all distinct corranged minors of given order 61
Derived sequences, Theorems concerning 62—71
Determinant, Agreement with usual definition 50
Determinant, Bordered detant, expansion of 240
Determinant, Conjugate detant, self-conjugate or symmetrical detant 5
Determinant, Conjugate reciprocal detant 111 180
Determinant, defined to be the detoid of a square matrix 12 15
Determinant, Detant of a product of three matrices whose two activities are equal 236 (see also “Determinoid”)
Determinant, Detant of any matrix product whose two activities are equal 220 232 236
Determinant, Eeciprocal detant 110 267
Determinant, Leading diagonal, order, as in square matrix 1
Determination of complete sets of 388 397 399
Determinoid of a product of two matrices 209—20
Determinoid of a product of two matrices when both matrices are square 210
Determinoid of a product of two matrices when passivity = efficiency 213
Determinoid of a product of two matrices when passivity efficiency 215—7
Determinoid of a product of two matrices when passivity<efficiency 209 212
Determinoid of a product of two matrices, General formula for 210
Determinoid of a product of two matrices, Rules for evaluation of 217
Determinoid of a product of two matrices, Special cases one matrix a scalar matrix 219
Determinoid of a product of two matrices, Special cases one matrix a unit matrix 219
Determinoid of a product of two matrices, Special cases two inversely similar matrices 220
Determinoid of a rectangular matrix, Abbreviated notations for 17—19 347
Determinoid of a rectangular matrix, Buie of signs for 12
Determinoid of a rectangular matrix, defined as sum of affected complete derived products of the matrix 24
Determinoid of any matrix product 221—32
Determinoid of any matrix product (cont.), Special cases, one factor a scalar matrix 231
Determinoid of any matrix product (cont.), Special cases, one factor a unit matrix 231
Determinoid of any matrix product (cont.), Special cases, one factor numerical 231
Determinoid of any matrix product when no passivity < efficiency 225
Determinoid of any matrix product when one passivity < efficiency 221
Determinoid of any matrix product, Expressions for, when the activities are equal 232 236
Determinoid of any matrix product, Expressions for, when they are unequal 243
Determinoid of any matrix product, General formulae for 227 228
Determinoid of any matrix product, Progressive development of 229
Determinoid of any matrix product, Theorems concerning 221
Determinoid, definitions of terms used in connection with, Affects, corranged and deranged minors, derangements, derived detoids, derived matrices, derived products, leading element, leading line, leading position, leading rows, long and short rows, minor detoids, minor matrices, moves, orders, rank, steps have the same signification for the determinoid of a matrix as for the matrix itself (see “Rectangular matrix Affects”)
Determinoid, generalisation of 48 134 136—7 244
Determinoid, properties of 19—21 36 43 48 52—4 91—104 105-52
Determinoid, properties of, Properties of, inferior minors 116—23 134—40
Determinoid, properties of, Properties of, long rows 19—21 36 43—4 105-9
Determinoid, properties of, Properties of, long-cut minors 116—23 136-8
Determinoid, properties of, Properties of, short rows 111—3 140
Determinoid, properties of, Properties of, short-cut minors 113—5 123—7 134—6 138-40
Determinoid, properties of, Properties of, superior minors 113—5 123-7
Determinoid, use of terms conjugate and reciprocal, A detoid and its conjugate have equal values 17
Determinoid, use of terms conjugate and reciprocal, The conjugate detoid of a matrix is the detoid of the conjugate matrix 5
Determinoid, use of terms conjugate and reciprocal, The conjugate reciprocal detoid of a matrix is the detoid of the conjugate reciprocal matrix 111
Determinoid, use of terms conjugate and reciprocal, The reciprocal detoid of a matrix is the detoid of the reciprocal matrix 110
Determinoids, Addition of 107
Determinoids, Addition of, Addition to one long row of scalar multiples of other long rows 107—9
Determinoids, Addition of, Affected minor detant invariant for all derangements of its rows 101
Determinoids, Addition of, Affected minor detoid invariant for all derangements of its long rows 101
Determinoids, Addition of, Bordered detoid 241
Determinoids, Addition of, Interchange of two long rows 20 36 43
Determinoids, Addition of, Multiplication by a scalar number 19 108
Determinoids, Addition of, Reciprocal of a detoid whose orders differ by 1 268
Determinoids, Addition of, Reversals of rows 53
Determinoids, Addition of, Two long rows identical 20 44
Determinoids, Addition of, Vanishing of a long row 107
Determinoids, expansions of, in terms of simple minor detants 113
Determinoids, expansions of, in terms of simple minor detants, generalisation 123
Determinoids, expansions of, in terms of the elements of a long row 105
Determinoids, expansions of, in terms of the elements of a long row, generalisation 110
Determinoids, expansions of, sin terms of the simple minor detants of a long-cut minor matrix 116
Determinoids, expansions of, sin terms of the simple minor detants of a long-cut minor matrix, generalisations 121—3 143
Dissimilar matrices, Addition and subtraction of 156
Dissimilar matrices, Equality of 154
Distinct minors of a sequence 232 248
Distinct minors of a sequence, horizontal and vertical minor matrices 249
Distinct minors of a sequence, minor detants 115 249
Distinct minors of a sequence, standard arrangement of 251
Distributive law for matrix products 173 184 188—9
Distributive law for multiplication by a scalar number 157
Double-suffix notation for detoids 17 18
Double-suffix notation for matrices 5—9
Efficiency of a matrix (or detoid) 1
Efficiency of a matrix (or detoid) of a matrix product 191
Elements of a matrix (or detoid) 1
Elements of a matrix (or detoid), Affects of the elements, affected and unaffected elements vi
Elements of a matrix (or detoid), Expressions for the elements of a product matrix 159 162—3 194
Elements of a matrix (or detoid), Leading element 1
Elements of a matrix (or detoid), relative affects 24
Elements of a sequence; affects of the elements, relative affects, leading element 55—6
Equality of matrices 153—5
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