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



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Íàçâàíèå: Matrices and Determinoids

Àâòîð: Cullis C.E.

ßçûê: en

Ðóáðèêà: Ìàòåìàòèêà/

Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö

ed2k: ed2k stats

Ãîä èçäàíèÿ: 1913

Êîëè÷åñòâî ñòðàíèö: 430

Äîáàâëåíà â êàòàëîã: 25.03.2006

Îïåðàöèè: Ïîëîæèòü íà ïîëêó | Ñêîïèðîâàòü ññûëêó äëÿ ôîðóìà | Ñêîïèðîâàòü ID
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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 $Q^{n}_{m}$      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 $\textit{r}$      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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