Rao C.R., Mitra S.K. — Generalized inverse of matrices and its applications :: Электронная библиотека попечительского совета мехмата МГУ

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Rao C.R., Mitra S.K. — Generalized inverse of matrices and its applications

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Название: Generalized inverse of matrices and its applications

Авторы: Rao C.R., Mitra S.K.

Аннотация:

This book is an attempt to bring together all the available results on "invcrtibility of singular matrices" under a unified theory and \v discuss their applications. It is well known that if A is a square non-singular matrix, then there exists a matrix G, such that AG = GA * I, which is called the inverse of A and denoted by A "1 If A is a singular or a rectangular matrix, nc such matrix G exists.

Язык:

Рубрика: Математика/

Серия: Сделано в холле

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1972

Количество страниц: 256

Добавлена в каталог: 17.03.2011

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

${Tr}_{(i)}{A}$       8 81
Adjoint matrix      4
Admittance matrix      181
Admittance-impedance conversion      185
Anderson, W.N., Jr.      188 189
Basis of a vector space      2
Basis of a vector space, dual basis      2
Basis of a vector space, reciprocal basis      2
Ben — Israel, A.      119 192 194 211
Best approximate solution      61 68
Best linear minimum bias estimator (BLMBE)      139
Best linear unbiased estimator (BLUE)      139
Bhimasankaram, P.      42 132
Bidiagonalization      5
BjBrck, A.      211
Bjerhammer, A.      16
Blattner, J.W.      69
Bleick, W.E.      153
Bott, R.      17 99
Businger, P.      211
Cayley — Hamilton theorem      8
Characteristic equation      8
Characteristic function (polynomial)      7
Characteristic function (polynomial), computation of      81
Charnes, A.      192 194 195
Chernoff, H.      203
Chipman, J.S.      138
Chisquare distribution, central      168
Chisquare distribution, noncentral      168
Cline, R.E.      16 67 71 98
Cochran's Theorem, generalization of      174
Cochran, W.G.      178
Cogredient transformation      121
Commuting hermitian matrices, transformation of      124
Commuting inverse      17 79
Commuting inverse, quasi commuting      93
Companion matrix      9 84
Computation of g-inverse      42 88 207
Conjugate transpose      12
Constrained inverse      17 98
Contragredient transformation      121
Convex programming      195
Craig — Sakamoto theorem      170
Craig, A.T.      170 178
Cyclic decomposition of a vector space      82
Cyclic spaces (types I and II)      83
Decell, H.P., Jr.      69
Decomposition theorem for matrices      33 93
Direct sum, matrices      10
Direct sum, vector spaces      3
Discriminant function      203
Disjoint (virtually) subspaccs      3
Drazin, M.P.      17 95
Duality theorem      50
Duffin, R.J.      17 99 188 189
Eigen values and vectors      8 86
Eigen values and vectors, $\rho -$ and $\chi$ -invers      74
Eigen values and vectors, generalized eigen value equation      43
Eigen values and vectors, generalized eigen vector      74
Eigen values and vectors, generalized null vector      89
Eigen values and vectors, inverse commuting with a power of the matrix      92
Eigen values and vectors, inverse with eigen value property      74 82 103
Eigen values and vectors, inverse with power property      78
Eigen values and vectors, of maximal height      89
Eigen values and vectors, parallel sum      191
Eigen values and vectors, proper eigen values      125
Eigen values and vectors, square partial isometry      116
Elementary divisors of a pencil      1 29
Englefield, M.J.      17 77
Equivalence of matrices      8
Erdelyi, I.      16 18 92 114
Faddeev, D.K.      81
Fisher — Cochran theorem      174
Frame, J.S.      16 28
Francis, J.      211
Frobenius inequality      18
g-Inverse, classification of, $\chi$ -inverse      15 16 73
g-Inverse, classification of, $\rho *\chi$ -inverse      16 97
g-Inverse, classification of, $\rho$ -inverse      15 16 75
g-Inverse, classification of, $\rho\chi *$ -inverse      16 97
g-Inverse, classification of, $\rho\chi$ -inverse      15 16 75 76 79 80
g-Inverse, classification of, basic solution      14 17 29
g-Inverse, classification of, commuting inverse      17 79
g-Inverse, classification of, commuting pseudoinverse      17 95
g-Inverse, classification of, constrained inverse      17 98
g-Inverse, classification of, general reciprocal      16
g-Inverse, classification of, group inverse      16
g-Inverse, classification of, least squares      14 16 48
g-Inverse, classification of, least squares, M-least squares      16 49
g-Inverse, classification of, left inverse      14 16 19
g-Inverse, classification of, left power inverse      16 98
g-Inverse, classification of, maximum rank      14 17 33
g-Inverse, classification of, minimum N-norm      14 45
g-Inverse, classification of, minimum norm      14 16 44
g-Inverse, classification of, minimum norm least squares      14 16 50
g-Inverse, classification of, Moore Penrose      16
g-Inverse, classification of, normalized generalized      16
g-Inverse, classification of, oblique      105
g-Inverse, classification of, pseudoinverse      16
g-Inverse, classification of, quasi-commuting      93
g-Inverse, classification of, reciprocal inverse      16
g-Inverse, classification of, reflexive inverse      14 16 27 68
g-Inverse, classification of, restricted      104
g-Inverse, classification of, right inverse      14 16 19
g-Inverse, classification of, right power inverse      16 98
g-Inverse, classification of, semi-inverse      16
g-Inverse, classification of, specified rank      31
g-Inverse, computation of      42 88 207
g-Inverse, definition      20 21
g-Inverse, expressions for different types      207
g-Inverse, representation of      26
g-Inverse, with specified properties, eigen value property      17 74 82 103
g-Inverse, with specified properties, from Jordan decomposition      17 87
g-Inverse, with specified properties, in the subalgebra of the matrix      80
g-Inverse, with specified properties, power property      77
g-Inverse, with specified properties, specified manifolds      72
Gantmacher, F.R.      1 128
Gauss — Markov model      136
Gauss — Markov model, addition or removal of observation      150
Gauss — Markov model, minimum bias estimator      138
Gauss — Markov model, multivariate case      154
Gauss — Markov model, optimality of least squares estimator      155
Gauss — Markov model, simultaneous estimation      141
Gauss — Markov model, singular covariance matrix      148
Gauss — Markov model, tests of linear hypothesis      142
Gauss — Markov model, unbiased estimator      137
Gauss — Markov model, validity of SLE for deviations in design & dispersion matrix      16
Gauss — Markov model, validity of SLE for deviations in design matrix      162
Gauss — Markov model, validity of SLE for deviations in dispersion matrix      155
Generalized eigen vector      see “Eigen values and vectors”
Goldman, A.J.      16
Golub, G.      211 216
Graybill, F.A.      178
Greville, T.N.E.      68 69
Hadamard product      11
Henderson, C.R.      197
Hermite canonical form      4 18 33 40 209
Hogg, R.V.      178
Householder transformation      5 215
Householder, A.S.      1
Hypercompanion matrix      10
Idempotent matrix      13 23 31 111 118 119
Idempotents of a g-inverse      29
Ijiri, Y.      70
Impedance coefficients      184
Indefinite admittance matrix      181
Index (Drazin) of a matrix      91 96 97
Invariant factors      9
Jordan canonical form      10
Katz, I.J.      18

Khatri — Rao product      12
Khatri, C.G.      12 41 70 111 178 204
Kirby, M.      195
Kronecker product      11 25 60
Kruskal, W.      157
Least squares inverse      14 16 48
Least squares, addition/removal of observation      150
Least squares, normal equations, solution of      213
Least squares, theory      140
Left inverse      14 16 19
Leverrier — Fadeev method      81
Linear programming, g-inverse for basic solution      29
Linear programming, interval      192
Linear programming, nonlinear convex programming      195
Marsaglia, G      178
Matrices of index      1 97
Matrices of index, equivalent conditions      18 71
Matrix equation      55
Matrix equation, best approximate solution of      61 68
Matrix types, EP      13 18
Matrix types, EPr      13 117 119
Matrix types, hermitian      13
Matrix types, idempotent      13 23 31 111 118 119
Matrix types, involution      118
Matrix types, nilpotent      13 93
Matrix types, nonnegative definite      13 41
Matrix types, normal      6 13 67 103 119
Matrix types, orthogonal      13
Matrix types, orthogonal complement      12
Matrix types, positive definite      13
Matrix types, positive semidefinite      13
Matrix types, semisimple      10 13 42
Matrix types, semiunitary      116
Matrix types, subunitary      114
Matrix types, symmteric      13
Matrix types, tripotent      113
Matrix types, unitary      6 13 114
Maximum Likelihood Estimation      201
Milne, R.D.      105
Minimal indices of a pencil      129
Minimum bias and variance estimation      139
Minimum bias estimator      138
Minimum norm (semi-norm) inverse      14 16 44
Minimum norm least squares inverse      14 16 50
Minimum polynomial      8 9 80
Minimum variance estimator      139
Mitra, S.K.      16 200 203
Mitra, Sanjit, K.      184
Moore — Penrose inverse ${A}^{+}$       14 16 51
Moore — Penrose inverse ${A}^{+}$ , expansion of      61
Moore — Penrose inverse ${A}^{+}$ , expressions for      52 53 69
Moore — Penrose inverse ${A}^{+}$ , incidence matrix      70
Moore — Penrose inverse ${A}^{+}$ , Lagrange — Sylvester formula      61
Moore — Penrose inverse ${A}^{+}$ , Neumann type expansion      62
Moore — Penrose inverse ${A}^{+}$ , partitioned matrix      71
Moore — Penrose inverse ${A}^{+}$ , product of matrices      67
Moore, E.H.      16 50 101
Network theory, application of g-inverse, admittance impedance conversion      185
Network theory, impedance coefficients      184
Network theory, indefinite admittance matrix      181
Network theory, parallel sum of matrices (1)      186
Network theory, parallel sum of matrices (2)      188
Network theory, reduction of a multipole      183
Nilpotent matrix      13 35 93 104
Normal matrix      6 13 67 103
Null space      105
Oblique inverse      1C5
Odell, P.L.      17 76 89
Ogasawara, T.      172
Ogawa, J.      178
Orthogonal projector      108 111
Orthogonal subspace      3
Orthogonalization r.ieinod      210
Parallel sum of matrices      12 186 188
Partial isometry      3 114
Partial isometry, normal      119
Partial isometry, supplements of      116
Partitioned matrix, g-inverse of      41 64
Pearl, M.H.      18
Pease, M. C      1
Pedis, S.      1
Penrose, R.      16 51 68
Polar reduction of a matrix      7
Polar representatkn of a matrix      7
Power property, g-ir verse with      77
Principal idempoteits      35 104
Product of matrices, Hadamard      11
Product of matrices, Khatri — Rao      1 2
Product of matrices, Kronecker      11 25 60
Projection operator      3 106
Pyle, L.D.      210
Q-R algorithm      211
Quadratic forms, condition for independence      177
Quadratic forms, correlated case      171
Quadratic forms, distribution of      168
Quadratic forms, n.s. conditions for reducibility      121
Quadratic forms, pair of forms      120
Quadratic forms, reduction of      1 20
Quadratic forms, several forms      131
Quadratic forms, uncorrelated case      169
Rank factorization      5 21 28 209
Rao, C.R.      1 4 12 16 47 158 162 168 175 178 199 202 203 206 210 211
Reciprocal (dual) basis      2
Reduction of matrices, bidiagonalization      5
Reduction of matrices, diagonal      5 209
Reduction of matrices, hermite canonical form      4 18 33 40 209
Reduction of matrices, polar      7
Reduction of matrices, singular value      6 38 53 62 209 211
Reduction of matrices, spectral decomposition      5 38
Reduction of matrices, triangular      5 215
Reed, M.B.      186
Reflexive inverse      14 16 27 68
Reinsch.C      211 216
Right inverse      14 16 19
Robers, P.D.      194
Rohde, C.A.      16 41 103 197 201
Rosen, J.B.      17 30
Sakamoto, H.      170 178
Schur's lemma      12
Scroggs, J.E.      17 76 89
Searle, S.R.      197
Seminorm      3 46 49 51
Semisimple matrix      10 13 42
Semiunitary matrix      116
Seshu, S.      186
Shanbag, D.N.      172
Sharpe, G.E.      182 184
Sheffield, R.D.      16
Sibuya, M.      140
Similar matrices      9
Similarity invariants      9
Singular normal distribution      203
Singular value decomposition      6 38 53 62 209 211
Singular value decomposition, for two matrices      6
Singular value decomposition, MN decomposition      7
Singular values and vectors      12
Smith's canonical form      8
Solution of equations, ${(XX*)}^{-}X = M      60
Solution of equations, ${{\Sigma}_{i}{a}_{i}(XX*)}^{\rho -i}X = M$       60
Solution of equations, A*XA = 0      25
Solution of equations, AX = 0      24
Solution of equations, AX = C, XB = D      25
Solution of equations, AX = Y      24 27
Solution of equations, AXB = C      24
Solution of equations, BXB = B, (BX)* = BX      57
Solution of equations, BXBXB = BXB      58
Solution of equations, XBX = 0      57
Solution of equations, XBX = 0, WBX = 0      57
Solution of equations, XBX = X      55
Solution of equations, XBX = X, (BX)* = BX      56

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