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Wan Z.-X. — Geometry of matrices
Wan Z.-X. — Geometry of matrices

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Название: Geometry of matrices

Автор: Wan Z.-X.

Аннотация:

The present monograph is a state of the art survey of the geometry of matrices. Professor L. K. Hua initiated the work in this area in the middle forties. In this geometry, the points of the space are a certain kind of matrices of a given size, and the four kinds of matrices studied by Hua are rectangular matrices, symmetric matrices, skew-symmetric matrices and hermitian matrices. To each such space there is associated a group of motions, and the aim of the study is then to characterize the group of motions in the space by as few geometric invariants as possible. At first, Professor Hua, relating to his study of the theory of functions of several complex variables, began studying the geometry of matrices of various types over the complex field. Later, he extended his results to the case when the basic field is not necessarily commutative, discovered that the invariant "adjacency" alone is sufficient to characterize the group of motions of the space, and applied his results to some problems in algebra and geometry. Professor Hua's pioneer work in the area has been followed by many mathematicians, and more general results have been obtained. I think it is now time to summarize all results obtained so far, and this has been my motivation for the present work.


Язык: en

Рубрика: Математика/

Серия: Сделано в холле

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1996

Количество страниц: 381

Добавлена в каталог: 30.10.2011

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
A system of coordinates      67
Action of a group      28
Addition of matrices      13
Addition of vectors      1 9 10
Adjacence      90 124 153 154 157 182 198 203 210 211 217 288 296 302 306 358 363 364 365
Affine equivalence      51
Affine geometry      51
Affine group      51
Affine r-flat      45
Affine transformation      51
Alternate matrix      39
Anti-automorphism of a division ring      17 35
Anti-isomorphism of division rings      66 80
Arithmetic distance      89 124 157 182 217 288 306 358
Automorphism of a division ring      55 78
Basis      2 3 10
Block form of a matrix      18
Block of a matrix      18
Center      36 71 306
Cogredience transformation      36 39
Coherence      90 124
Column rank      21
Column vector      9 12
Column vector space      9
Connected graph      153
Consistent system of linear equations      34 47
Coordinate(s)      45 67
Diagonal matrix      16
Diameter      153
Dickson invariant      200
Dimension formula      7 50 69
Dimension of a subspace      6
Dimension of a vector space      10
Dimension of an r-flat      45 68
Distance      91 127 159 184 203 219 289 308 359
Distance-transitive graph      153
Division ring      1
Dual flat      75 178 286 357
Dual polar graph of type $C_{n}$      302
Dual polar graph of type $D_{n}$      212
Dual polar graph of type ${}^{2}A_{2n-1}$      364
Dual polar space of type $B_{n}$      3
Dual polar space of type $C_{n}$      288
Dual polar space of type ${}^{2}A_{2n-1}$      358
Dual polar space of type ${}^{2}A_{2n}$      363
Dual polar space of type ${}^{2}D_{n+1}$      181
Dual subspace      33 178 286 356
Echelon matrix      26
Echelon normal form      26
Equation of a hyperplane      47 68
Equivalence of matrices      23
Erlangen program      51 71
Finite dimensional left vector space      10
Finite dimensional right vector space      11
Finite point      74 80 124 181 288 358
Fixed field      36 306
Fundamental theorem of the affine geometry      54
Fundamental theorem of the geometry of alternate matrices      158
Fundamental theorem of the geometry of hermitian matrices      307
Fundamental theorem of the geometry of rectangular matrices      90
Fundamental theorem of the geometry of symmetric matrices      218
Fundamental theorem of the irreducible space      204
Fundamental theorem of the one-dimensional projective geometry      85
Fundamental theorem of the projective geometry      77
Fundamental theorem of the projective geometry of $m \times (m + n)$ matrices      143
Fundamental theorem of the projective geometry of alternate matrices      185
Fundamental theorem of the projective geometry of hermitian matrices      360
Fundamental theorem of the projective geometry of symmetric matrices      290
General linear group      20
Generalized orthogonal group      185 198 211
Generalized orthogonal matrix      185 211
Generalized projective transformation      84
Generalized Symplectic group      289
Generalized symplectic matrix      289
Generalized unitary group      360 362
Generalized unitary matrix      359
Generator      3
Graph      153
Graph automorphism      153
Graph isomorphism      153
Graph of hermitian matrices      363
Graph of symmetric matrices      296
Graph of the irreducible space      213
Graph of the left Grassmann space      154
Graph of the projective space of $n \times n$ alternate matrices      212
Graph of the projective space of $n \times n$ hermitian matrices      364
Graph ofmxn matrices      153
Graph ofnxn alternate matrices      211
Group of motions      89 124 157 217 306
Half dual polar graph of type $D_{n}$      213
Harmonic set      81
Hermitian matrix      35
Homogeneous coordinate(s)      74 124 181 288 358
Hyperplane      45 66
Hyperplane at infinity      74
Idempotent matrix      119 283
Identity matrix      16
Improper generalized orthogonal matrix      204 211
Improper orthogonal matrix      199 200 210
Incidence relation      45 66
Independent set of points      69
Independent system of linear equations      34 47
INDEX      177 179
Intersection of flats      50 68
Intersection of subspaces      7
Inverse matrix      19
Invertible matrix      19
Irreducible space      201 210
Isomorphism of division rings      66 80
Isomorphism of Jordan rings      122 281
Isomorphism of vector spaces      11
Join of flats      50 68
Jordan isomorphism      122
Jordan multiplication      122
Jordan ring      122 281 355
Left affine space      45
Left Grassmann space      123
Left inverse      20
Left invertible matrix      20
Left projective line      80
Left projective space      67
Left vector space      1 10
Line      45 66 95 130 165 230 251 320 322 323 345 347 348
linear combination      2 3
Linear dependence      2
Linear independence      2
Matrix      12
Matrix representation of a subspace      28
Maximal set      159
Maximal set of rank      2 97 224 244 270 311
Maximal totally isotropic subspace      179 287 357 362
Maximal totally singular subspace      180 198
Method of intersection and join      60
Modular law      8
Multiplication formula of matrices in block form      19
Multiplication of matrices      15
Multiplicator      185
Negative of a matrix      13
Non-homogeneous coordinate(s)      74 80 124 144 181 187 288 291 358 361
Nonalternate symmetric matrix      41
Nonzero subspace      3
Norm map      305
Orbit      29
Orthogonal group      178 180 198
Orthogonal idempotent matrices      119 283
Orthogonal matrix      177 179
Orthogonal polarity      178
Parallel flats      49
Parametric equation      95 104
Parametric representation      49 70
Perfect field      300
Permutation matrix      23
Plane      45 66 104 167
Point      45 66 89 123 124 157 181 198 217 306 358 363
Point at infinity      74 80 124 181 288 358
Primitive idempotent matrix      119 283
Principal diagonal      16
Principle of duality      75
Product of matrices      15
Projective equivalence      71
Projective general linear group      71
Projective geometry      71
Projective line      80
Projective r-flat      66
Projective space of $(m + n) \times n$ matrices      138
Projective space of $m \times (m + n)$ matrices      124
Projective space ofnxn alternate matrices      181
Projective space ofnxn hermitian matrices      358
Projective space ofnxn symmetric matrices      287
Projective transformation      70 71
Proper generalized orthogonal group      204 211
Proper generalized orthogonal matrix      204 211
Proper orthogonal group      200 210
Proper orthogonal matrix      199 200 210
r-flat      45 66
Rank      24
Reduced maximal set of rank      2 318
Replacement Theorem      4
Right affine space      54
Right Grassmann space      138
Right inverse      20
Right invertible matrix      20
Right projective space      75
Right vector space      9
Row equivalence of matrices      26
Row rank      21
Row vector      1 12
Row vector space      1
scalar      10 14
Scalar multiplication of a matrix by a scalar      14 15
Scalar multiplication of a vector by a scalar      1 9 10
Scalar product of a matrix by a scalar      14
Scalar product of a vector by a scalar      10
Self-dual flat      179 357
Self-dual subspace      179 287 357
Semi-automorphism of a division ring      82
Semi-isomorphism of division rings      82
Semi-isomorphism of rings      120
Skew lines      49
Skew-symmetric matrix      39
Solution      30
Solution space      31
Solution vector      30
Space of $m \times n$ matrices      89
Space of $n \times n$ alternate matrices      157
Space of $n \times n$ hermitian matrices      306
Space of $n \times n$ symmetric matrices      217
Space of rectangular matrices      89
SubMatrix      17
subspace      3
Sum of matrices      13
Sum of subspaces      7
Sum of vectors      10
Symmetric matrix      37
Symmetrized multiplication      122 281
Symplectic group      286
Symplectic matrix      285
Symplectic Polarity      286
System of linear equations      29
System of linear homogeneous equations      30
System of linear non-homogeneous equations      30
Total matrix ring      118
Totally isotropic subspace      179 198 287 357 362
Totally singular subspace      180 198
Trace map      36 305
Transitive set      29
Transpose of a matrix      17
Unitary group      356 362
Unitary matrix      356
Unitary polarity      357
Vector      10
Vector space      1
Vertex      153 154 211 296 302 363 364 365
Zero matrix      13
Zero subspace      3
Zero vector      2 10
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