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| Cullis C.E. — Matrices and Determinoids |
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| Ïðåäìåòíûé óêàçàòåëü |
Passive rows in a product of two matrices 158
Passive rows in any matrix product 190
Passivities of a matrix product, and of matrix factors 158 191
Passivities of a matrix product, and of matrix factors, Conditions for cancellation 357
Passivities of a matrix product, and of matrix factors, Conditions for non-zero solutions 325 336 354 381
Post-factors, applied to solve a system of unconnected linear equations 371—2
Post-factors, applied to solve the irreducible matrix equations XB=C, AXB=C 333 352
Post-factors, inverse, of a matrix 322
Pre-factors, applied to solve a system of unconnected linear equations 371
Pre-factors, applied to solve the irreducible matrix equations AX=C, AXB=C 323 352
Pre-factors, inverse of a matrix 335
Principal inverse of a matrix 182 (see “Inverse”)
Product formed by a chain of matrix factors.— Special eases, a scalar matrix 207 231
Product formed by a chain of matrix factors.— Special eases, a unit matrix 207 231
Product formed by a chain of matrix factors.— Special eases, inferior minors 116—23 134—40
Product formed by a chain of matrix factors.— Special eases, long rows 19—21 36 43—4 105-9
Product formed by a chain of matrix factors.— Special eases, long-cut minors 116—23 136-8
Product formed by a chain of matrix factors.— Special eases, One factor a zero matrix 207
Product formed by a chain of matrix factors.— Special eases, passive rows 164—8 195—9 217 222
Product formed by a chain of matrix factors.— Special eases, Products, derived 22—52
Product formed by a chain of matrix factors.— Special eases, Properties of active rows 168—72 199—204 217 222-3
Product formed by a chain of matrix factors.— Special eases, short rows 111—3 140
Product formed by a chain of matrix factors.— Special eases, short-cut minors 113—5 123—7 134—6 138-40
Product formed by a chain of matrix factors.— Special eases, Standard product of square matrices, reciprocal and conjugate reciprocal of 261
Product formed by a chain of matrix factors.— Special eases, superior minors 113—5 123-7
Product formed by a chain of matrix factors.— Special eases, the number  127—34
Product formed by any chain of matrix factors 188—208 221—36 243—4 244—7 253-60
Product formed by any chain of matrix factors is associative 188—9
Product formed by any chain of matrix factors is distributive 189—90
Product formed by any chain of matrix factors is in general not commutative 189 174 208
Product formed by any chain of matrix factors, Active rows 191 199—204 222-3
Product formed by any chain of matrix factors, Activities, defined 191
Product formed by any chain of matrix factors, Conjugate of 205
Product formed by any chain of matrix factors, Conjugate reciprocal 261
Product formed by any chain of matrix factors, Corresponding rows 191—2
Product formed by any chain of matrix factors, Detoid of 221 236
Product formed by any chain of matrix factors, Efficiency, defined 191
Product formed by any chain of matrix factors, Elements of product matrix, expressions for 194—5
Product formed by any chain of matrix factors, Equivalence of conventionally equal matrices 204—5
Product formed by any chain of matrix factors, Generalisation of detoid 253—60
Product formed by any chain of matrix factors, Multiplication by a scalar number 206
Product formed by any chain of matrix factors, Partial products, product expressed as a sum of 195—9 222
Product formed by any chain of matrix factors, Passive rows 191 195—9 222
Product formed by any chain of matrix factors, Passivities, defined 191
Product formed by any chain of matrix factors, Product matrix, form of 190
Product formed by any chain of matrix factors, Properties of 195—9 222
Product formed by any chain of matrix factors, Rank of product, superior limit to 283
Product formed by any chain of matrix factors, Reciprocal and conjugate reciprocal of a standard product of square matrices 261
Product formed by any chain of matrix factors, Reduction to 192
Product formed by any chain of matrix factors, Standard form of product 192
Product matrix, Conjugate of 175 205
Product matrix, defined 158 161 187 189
Product matrix, Expressions for elements of 159 162—3 194-5
Product matrix, Form of 159 163 190
Product matrix, Reciprocal and conjugate reciprocal of 261
Product of a matrix and a scalar number 157
Product of a matrix and a scalar number is commutative and distributive 157
Product of three matrices 184—8 236—43
Product of three matrices is associative 186
Product of three matrices is distributive 188
Product of three matrices is in general not commutative 174 208
Product of two matrices 158—84 209—20
Product of two matrices is always distributive 173
Product of two matrices is in general not commutative 174
Product of two matrices, Active rows 158 168—72 217
Product of two matrices, Activities, defined 158
Product of two matrices, Commutative products 175 181—3 262-3
Product of two matrices, Conjugate of 175
Product of two matrices, Conjugate reciprocal 261
Product of two matrices, Corresponding rows 160 163—4
Product of two matrices, Detoid of 209—20
Product of two matrices, Efficiency, defined 191
Product of two matrices, Elements of product matrix, expressions for 159 162—3
Product of two matrices, Passive rows 158 191 164—8 217
Product of two matrices, Passivities, defined 158
Product of two matrices, Product matrix, form of 159 163
Product of two matrices, Properties of 164—8 217
Product of two matrices, Rank of product matrix 281—4 288—9 293-4
Product of two matrices, Reciprocal and conjugate reciprocal of a standard product of two square matrices 261
Product of two matrices, Reduction to 162
Product of two matrices, Redundant final passive rows 161 165
Product of two matrices, Standard form 158
Product of two matrices.—Special cases, A matrix and its conjugate reciprocal 180
Product of two matrices.—Special cases, An undegenerate matrix and its principal inverse 181
Product of two matrices.—Special cases, Commutative products, A square matrix and its conjugate reciprocal 181 263 309
Product of two matrices.—Special cases, Commutative products, An undegenerate square matrix and its inverse 183 309
Product of two matrices.—Special cases, One factor a unit matrix 177
Product of two matrices.—Special cases, One factor a unit matrix an undegenerate square matrix 283
Product of two matrices.—Special cases, One factor a unit matrix, a quasi-scalar matrix 179
Product of two matrices.—Special cases, One factor a unit matrix, a scalar matrix 178
Product of two matrices.—Special cases, Two inversely similar matrices 183
Product of two matrices.—Special cases, Two mutually conjugate matrices 288—9 389
Product of two matrices.—Special cases, Two mutually inverse matrices 182—3 267
Product of two matrices.—Special cases, Two square matrices whose product is a non-zero scalar matrix 266 309
Properties of 164—8 195—9 222
Quasi-scalar matrix 179
Rank of a matrix, Conditions that rank shall be  279 288
Rank of a matrix, Criteria for rank 279 280 286 288
Rank of a matrix, defined 265
Rank of a matrix, Theorems concerning 278—89
Rank of a matrix, Theorems concerning connections and 271—8
Rank of a matrix, unaltered by derangements 266
Rank of a matrix, unaltered when the matrix is multiplied by or into an undegenerate square matrix 283
Rank of a matrix, Zero rank 265
Rank of a product of three matrices, Eank of any matrix product, cannot exceed rank of any factor 283
Rank of a product of three matrices, Equipotent matrices have equal ranks 285
Rank of a product of three matrices, Three undegenerate matrices with a common passivity 285
Rank of a product of two matrices 1
Rank of a product of two matrices, cannot exceed rank of either factor 281
Rank of a product of two matrices, Special eases, both factors are undegenerate and one has rank equal to its passivity 282 293
Rank of a product of two matrices, Special eases, one factor is an undegenerate square matrix 283
Rank of a product of two matrices, Special eases, two mutually conjugate real matrices 288
Rank of a product of two matrices, Special eases, two mutually conjugate undegenerate matrices 288—9
Rank of augmented matrix 287
Rank of augmented matrix, complete matrix of minor detants 289
Rank of augmented matrix, conjugate matrix 266
Rank of augmented matrix, functional matrix 294
Rank of augmented matrix, reciprocal of a square matrix 267 289
Rank of augmented matrix, solutions of the equations AX=0, XB=0, AXB=0 325 336 355
Rank of product of two mutually conjugate extravagant matrices 289
Rank of, in special eases 285
Ranks of two equipotent matrices 286
Real linear equations, systems of 387 389 399 406 411
Real linear equations, systems of, Complete sets of mutually orthogonal real unit solutions 406 411
Reciprocal matrix and detoid, conjugate reciprocal defined 111
Reciprocal matrix and detoid, defined 110
Reciprocal matrix and detoid, Fundamental property of the reciprocal or conjugate reciprocal matrix 110—1 180-1
Reciprocal matrix and detoid, Rank of the reciprocal of a square matrix 289
Reciprocal matrix and detoid, Reciprocal and conjugate reciprocal of a standard product of square matrices 261
Reciprocal matrix and detoid, Reciprocal detant of a square matrix 267
Reciprocal matrix and detoid, Reciprocal detoid of a matrix whose orders differ by 1 268
Reciprocal matrix and detoid, Reciprocal of a non-singular matrix is undegenerate 267
Reciprocal matrix and detoid, Reciprocal of an undegenerate square matrix is undegenerate 267
Rectangular matrix, Abbreviated notations 1 5—12 347
Rectangular matrix, defined 1
Rectangular matrix, definitions of terms used in connection with affects of derived or minor detoids and detants 86—8
Rectangular matrix, definitions of terms used in connection with affects of derived or minor matrices 86—8
Rectangular matrix, definitions of terms used in connection with affects of elements and derived products, vi 24
Rectangular matrix, definitions of terms used in connection with co-factor of a minor detant 119
Rectangular matrix, definitions of terms used in connection with co-factor of a minor detoid 141
Rectangular matrix, definitions of terms used in connection with co-factor of an element 106
Rectangular matrix, definitions of terms used in connection with complementary minor matrices or minor detoids 102
Rectangular matrix, definitions of terms used in connection with connections between the rows 269 295
Rectangular matrix, definitions of terms used in connection with derangements of the matrix 2
Rectangular matrix, definitions of terms used in connection with derived detoids and detants, minor detoids and detants 16
Rectangular matrix, definitions of terms used in connection with derived matrices 2
Rectangular matrix, definitions of terms used in connection with elements, horizontal and vertical orders, efficiency, long and short rows 1
Rectangular matrix, definitions of terms used in connection with leading element, leading line, leading row, leading position 1 2
Rectangular matrix, definitions of terms used in connection with minor matrices, corranged and deranged 2
Rectangular matrix, definitions of terms used in connection with moves: forward, backward, horizontal and vertical 48—9
| Rectangular matrix, definitions of terms used in connection with order of a derived product 4
Rectangular matrix, definitions of terms used in connection with rank of the matrix 265 295
Rectangular matrix, definitions of terms used in connection with simple minor matrices, horizontal and vertical 233—4 249
Rectangular matrix, definitions of terms used in connection with simple minor matrices, long-cut and short-cut 116
Rectangular matrix, definitions of terms used in connection with simple minor matrices, reduced and unreduced orders of 116
Rectangular matrix, definitions of terms used in connection with simple minor matrices, superior and inferior 116
Rectangular matrix, definitions of terms used in connection with steps: forward, backward, horizontal and vertical 4
Rectangular matrix, Standard notations for 1 5 10
Rectangular matrix.—Allied matrices defined, Complete matrices of minor detants of given order 248
Rectangular matrix.—Allied matrices defined, conjugate matrix 5
Rectangular matrix.—Allied matrices defined, Conjugate reciprocal matrix 111
Rectangular matrix.—Allied matrices defined, Inverse, principal inverse 182—4
Rectangular matrix.—Allied matrices defined, Reciprocal matrix 110
Rectangular matrix.—Allied matrices defined, Standard complete matrix of minor detants 251
Rectangular matrix.—Special forms, detant of 15
Rectangular matrix.—Special forms, leading diagonal 1
Rectangular matrix.—Special forms, order of 1
Rectangular matrix.—Special forms, Quasi-scalar matrix 179
Rectangular matrix.—Special forms, Scalar matrix 178
Rectangular matrix.—Special forms, Self-conjugate or symmetrical matrix 5
Rectangular matrix.—Special forms, Square matrix 1 15
Rectangular matrix.—Special forms, Unit matrix 11—12 177
Rectangular matrix.—Special forms, Zero matrix 155
Reduced order of a simple minor 116
Reduction of a matrix equation to an irreducible matrix equation 312 328 338
Reduction of a matrix product to standard form 162 184 192
Reduction of affects of derived products to affects of sequences 83
Reduction of any fundamental sequence to a sequence of natural numbers 60
Reduction to standard form 184—6 192
Redundant final passive rows in a matrix product 161 163 191
Relative affects of elements of a derived sequence 56
Relative affects of factors of a derived product 24
Reversals of rows in a detoid 53
Rows of 0’s, or zero rows, active, in a matrix product 200—1
Rows of 0’s, or zero rows, final, in any matrix 154
Rows of 0’s, or zero rows, long, in a detoid 107
Rows of 0’s, or zero rows, passive, in a matrix product 196—8
Rows of 0’s, or zero rows, short, in a detoid 112
Rows, Active and passive 158 190
Rows, Connected and unconnected 269 294
Rows, Corresponding, in a matrix product 160 163 191
Rows, Interchange of two parallel 30 38
Rows, Long and short 1 16
Rows, Properties of active 168 199 222
Rows, Properties of long 19—21 36 43—4 105-9
Rows, Properties of passive 164 195 222
Rows, Properties of short 111—3 140
Rows, Redundant final passive 161 163 191
Rows, Theorems concerning connections 271 287
Rows, two long rows 20 36 43
Rows, two short rows 44
Rule of Signs for a detoid 12
Rules for determining the affect of a derived product 25—30
Rules for determining the affect of a derived sequence 57—62
Rules for determining, change in affect of a derived sequence due to an interchange of elements 72 74 77 81
Rules for determining, changes in affect and sign of a derived product due to an interchange of rows or suffixes 30 36 38 44
Rules for evaluating the detoid of a product of two matrices 217
Rules for evaluating the detoid of any matrix product 222—5
Scalar matrix, defined, notation for 178
Scalar matrix, occurring as factor in a product of two matrices 178 219
Scalar matrix, occurring as factor in any matrix product 207 231
Scalar number, cannot be equatad to any non-zero matrix 179
Scalar number, Multiplication of a detoid by 19 108
Scalar number, Multiplication of a matrix by 157
Scalar number, occurring as a factor in any matrix product 206 231
Schemes of formation for complete matrices of the minor detants of order k of a given fundamental matrix 249—53
Schemes of formation for complete matrices of the minor detants of order k of a given fundamental matrix, Common schemes 250
Schemes of formation for complete matrices of the minor detants of order k of a given fundamental matrix, Corresponding schemes for the formation of a complete matrix of the minor detants of a matrix product 256
Schemes of formation for complete matrices of the minor detants of order k of a given fundamental matrix, Standard scheme 251
Self-conjugate matrix 5
Self-conjugate or symmetrical matrix, selfconjugate determinant 5
Short rows of 0’s 112
Short rows, defined 1 16
Short rows, Properties of 111—3 140
Short rows, Sum of products obtained by multiplying each element of a short row by its co-factor 140
Short-cut minors, defined 116
Short-cut minors, Properties of 113—5 123—7 134—6 138-40
Short-cut minors, Sums of products obtained by multiplying each simple minor detant of a given short-cut minor matrix by its co-factor 138
Short-cut minors, Sums of similar affected short-cut minor detoids 113 123 134 137
Sign determined by affect vi 24
Sign determined by affect, Change in sign of a derived product caused by an interchange of two rows or suffixes 30 36 38 44
Sign determined by affect, Interchange of two long rows changes the sign of a detoid 20 36 43
Sign determined by affect, Rule of Signs for a detoid 12
Sign determined by affect, Rule of Signs for a detoid, self-consistency of 13
Sign determined by affect, Sign of derived product independent of order of arrangement of factors 47 13 27
Sign determined by affect, Signs of all corranged minor sequences of given order 61
Similar matrices, Addition of 155
Similar matrices, defined 153
Similar matrices, Inversely similar matrices 153
Similar matrices, product of 179
Simple minor matrices and detoids, defined, classification 3 115
Simple minor matrices and detoids, Horizontal and vertical 233 234 249
Simple minor matrices and detoids, Inferior 116 134 136
Simple minor matrices and detoids, Long-cut 116 123 136
Simple minor matrices and detoids, Reduced and unreduced orders of 116
Simple minor matrices and detoids, Short-cut 116 123 134 138
Simple minor matrices and detoids, Sum s of similar affected simple minor detoids, inferior detoids 134 136
Simple minor matrices and detoids, Sum s of similar affected simple minor detoids, simple minor detants 115 209
Simple minor matrices and detoids, Sum s of similar affected simple minor detoids, superior detoids 123
Simple minor matrices and detoids, Superior 116 123
Single-suffix notations for detoids 18 19
Single-suffix notations for matrices 10
Single-suffix notations for sequences and one-rowed matrices 11 55 58
Singular and non-singular matrices, defined 266
Singular and non-singular matrices, Reciprocal, conjugate reciprocal and inverse of a non-singular matrix 267
Solutions of equations (see “Matrix equations” “Systems
Solvability of a system of linear algebraic equations 366 374
Solvability of a system of linear algebraic equations, Condition for non-zero solutions of a homogeneous system 381
Solvability of AX=0, XB=0, AXB=0, Conditions for non-zero solutions 325 336 354
Solvability of AX=C, XB=C, AXB=C, Conditions for finite solution 314 329 341
Solvability of AX=C, XB=C, AXB=C, Conditions for infinite solutions 327 337 357
Special forms, Quasi-scalar matrix 179
Special forms, Scalar matrix 178 207 231
Special forms, unit matrix 11 177 207 231
Special products of two undegenerate square matrices 181 183 262 266 309
Special products of undegenerate matrices 180—4 262 266—7 282 285 288 293 309
Square matrices, products of commutative products an undegenerate square matrix and its inverse 183 309
Square matrices, products of commutative products, a square matrix and its conjugate reciprocal 181 263 309
Square matrices, products of commutative products, Reciprocal and conjugate reciprocal of any standard product of square matrices 261
Square matrices, products of commutative products, two square matrices whose product is a non-zero scalar matrix 262 266 309
Square matrix, determinant of reciprocal 267
Square matrix, order, leading diagonal 1
Square matrix, rank of reciprocal 289
Standard arrangement of distinct minor sequences of given order 251
Standard complete matrix of minor detants 251 258
Standard equation for 258
Standard notations for matrix and detoid, double-suffix 1 5 17
Standard notations for matrix and detoid, single-suffix 10 18 11 55 58
Standard product, affects determined by counting, vi 24 12 13
Standard product, standard form of matrix product 158 184 192
Standard product, standard form of matrix product, expressions for elements of 195
Standard product, standard form of matrix product, reduction to 162 184 192
Standard product, steps in a matrix: forward, backward, horizontal and vertical 4
Standard schemes of formation 251 258
Steps in a sequence: forward and backward 55
Subtraction of matrices (see “Addition”)
Suffixes, Double-suffix notations 5—9 17 18
Suffixes, Single-suffix notations 10 18 19 11 55 58
Suffixes, vertical and horizontal 6 12
Sum of affected, (inferior) long-cut simple minor detoids of reduced order k 136 137
Sum of affected, complete derived products 12 27
Sum of affected, derived products of order k 134 136 137 244
Sum of affected, inferior short-cut simple minor detoids of reduced order k 134 137
Sum of affected, inferior simple minor detoids 134 136
Sum of affected, minor detants of order k 134 136—7 244
Sum of affected, simple minor detants 115 209
Sum of affected, superior short-cut simple minor detoids of reduced order k 123
Sum of affected, superior simple minor detoids 123
Sum of products obtained from a matrix or detoid by multiplying an element of a given long row by the co-factor of the corresponding element of another long row 107 108 110
Sum of products obtained from a matrix or detoid by multiplying, a simple minor detant of a given long-cut minor matrix by the co-factor of the corresponding simple minor detant of a similar long-cut minor matrix 122 123
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