Lectures on Matricies

Table of contents :
Chapter X. Linear Associative Algebras……Page 0
Title……Page 1
Preface……Page 2
Contents……Page 3
Corrigenda……Page 6
I. Linear transformations and vectors……Page 8
2. Linear dependence……Page 9
3. Linear vector functions and matrices……Page 10
4. Scalar matrices……Page 12
5. Powers of a matrix; adjoint matrices……Page 13
6. The transverse of a matrix……Page 15
8. Change of basis……Page 16
9. Reciprocal and orthogonal bases……Page 18
10. The rank of a matrix……Page 21
11. Linear dependence……Page 23
2. Matric polynomials in a scalar variable……Page 27
3-4. The division transformation……Page 28
5-6. The characteristic equation……Page 30
7-8. Matrices with distinct roots……Page 32
9-12. Matrices with mulitple roots……Page 34
13. The square root of a matrix……Page 37
14. Reducible matrices……Page 38
I. Elementary transformations……Page 40
2. The normal form of a matrix……Page 41
3. Determinantal and invariant factors……Page 43
4. Non-singular linear polynomials……Page 44
5. Elementary divisors……Page 45
6-7. Matrices with given elementary divisors……Page 46
8-9. Invariant vectors……Page 50
I. Vector polynomials……Page 54
2. The degree invariants……Page 55
3-4. Elementary sets……Page 56
5. Linear elementary bases……Page 59
6. Singular linear polynomials……Page 62
I. Automorphic transformation……Page 70
2-3. The equation y’ = +/-aya^-1……Page 71
4. Principal idempotent and nilpotent elements……Page 72
5. The exponential solution……Page 74
6. Matrices which admit a given transformation……Page 75
2. The scalar product……Page 77
3. Compound matrices……Page 78
5. Bordered determinants……Page 81
6-7. The reduction of bilinear forms……Page 82
8. Invariant factors……Page 85
9. Vector products……Page 86
10. The direct product……Page 88
11. Induced or power matrices……Page 89
12-14. Associated matrices……Page 90
15. Transformable systems……Page 93
16-17. Transformable linear sets……Page 94
18-19. Irreducible transformable sets……Page 99
I. Hermitian matrices……Page 102
2. The invariant vectors of a hermitian matrix……Page 104
3. Unitary and orthogonal matrices……Page 105
4. Hermitian and quasi-hermitian forms……Page 106
5. Reduction of a quasi-hermitian form……Page 107
6. The Kronecker method of reduction……Page 110
7. Cogredient transformation……Page 112
8. Real representation of a hermitian matrix……Page 114
I. Commutative matrices……Page 116
2. Commutative sets of matrices……Page 119
3. Rational methods……Page 120
4. The direct product……Page 122
5. Functions of commutative matrices……Page 124
6. Sylvester’s identities……Page 125
7. Similar matrices……Page 127
2. Infinite series……Page 129
3. The canonical form of a function……Page 130
4. Roots of 0 and 1……Page 132
5-6. The equation y^m = x; algebraic functions……Page 133
7. The exponential and logarithmic functions……Page 136
8. The canonical form of a matrix in a given field……Page 137
9. The absolute value of a matrix……Page 139
11. The absolute value of a tensor……Page 141
12. Matric functions of a scalar variable……Page 142
13. Functions of a variable vector……Page 144
14. Functions of a variable matrix……Page 149
15-16. Differentiation formulae……Page 150
I. Fields and algebras……Page 154
2. Algebras which have a finite basis……Page 155
3. The matric representation of an algebra……Page 156
4. The calculus of complexes……Page 157
5. The direct sum and product……Page 158
6. Invariant subalgebras……Page 159
7. Idempotent elements……Page 161
8-9. Matric subalgebras……Page 163
10-12. The classification of algebras……Page 165
13. Semi-invariant subalgebras……Page 170
14. The representation of a semi-simple algebra……Page 172
15. Group algebras……Page 174
Notes……Page 176
Bibliography……Page 179
Index to Bibliography……Page 201
Index……Page 204