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Cullis C.E. — Matrices and Determinoids
Cullis C.E. — Matrices and Determinoids



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Название: Matrices and Determinoids

Автор: Cullis C.E.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1913

Количество страниц: 430

Добавлена в каталог: 25.03.2006

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$Q^{n}_{m}$, defined      127
$Q^{n}_{m}$, Properties of      127—34
$Q^{n}_{m}$, Table of values      131
$Q^{n}_{m}=\sum(-1)^{\omega}$      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$\lessdot$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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