CHAPTER Ⅰ DEFINITIONS AND FUNDAMENTAL PROPERTIES OF MATRICES 1
1．Introductory 1
2．Definitions and Fundamental Properties 1
3．Matrix Multiplication 3
4．Reciprocal of a Non-Singular Matrix 4
5．The Reversal Law in Transposed and Reciprocal Products 4
6．Matrices Partitioned into Submatrices 5
7．Isolated Elements and Minors 8
8．Historical Note 9
CHAPTER Ⅱ ELEMENTARY TRANSFORMATIONS．BILINEAR AND QUADRATIC FORMS 10
1．The Solution of n Linear Equations in n Unknowns 10
2．Interchange of Rows and Columns in a Determinant or Matrix 10
3．Linear Combination of Rows or Columns in a Determinant or Matrix 12
4．Multiplication of Rows or Columns 12
5．Linear Transformation of Variables 13
6．Bilinear and Quadratic Forms 14
7．The Highest Common Factor of Two Polynomials 16
8．Historical Note 18
CHAPTER Ⅲ THE CANONICAL REDUCTION OF EQUIVALENT MATRICES 19
1．General Linear Transformation 19
2．Equivalent Matrices in a Field 19
3．The Canonical Form of Equivalent Matrices 20
4．Polynimials with Matrix Coefficients:λ-Matrices 21
5．The H．C．F．Process for Polynomials 22
6．Smith＇s Canonical Form for Equivalent Matrices 23
7．The H．C．F．of m-rowed Minors of a λ-Matrix 25
8．Equivalent λ-Matrices 26
9．Observations on the Theorems 27
10．The Singular Case of n Linear Equations in n Variables 29
11．Historical Note 30
CHAPTER Ⅳ SUBGROUPS OF THE GROUP OF EQUIVALENT TRANSFORMATIONS 32
1．Matrices of Special Type,Symmetric,Orthogonal,＆c． 32
2．Axisymmetric,Hermitian,Orthogonal,and Unitary Matrices 34
3．Special Subgroups of the Group of Equivalent Transformations 35
4．Quadratic and Bilinear Forms associated with the Subgroups 37
5．Geometrical Interpretation of the Collineation 40
6．The Poles and Latent Points of a Collineation 40
7．Change of Frame of Reference 41
8．Alternative Geometrical Interpretation 42
9．The Cayley-Hamilton Theorem 43
10．Historical Note 44
CHAPTER Ⅴ A RATIONAL CANONICAL FORM FOR THE COLLINEATORY GROUP 45
1．Linear Independence of Vectors in a Field 45
2．The Reduced Characteristic Function of a Vector 46
3．Fundamental Theorem of the Reduced Characteristic Function 47
4．A Rational Canonical Form for Collineatory Transformations 49
5．Properties of the R．C．F．＇s of the Canonical Vectors 52
6．Observations upon the Theorems 53
7．Geometrical and Dual Aspect of Theorem Ⅱ 53
8．The Invariant Factors of the Characteristic Matrix of B 54
9．Historical Note 56
CHAPTER Ⅵ THE CLASSICAL CANONICAL FORM FOR THE COLLINEATORY GROUP 58
1．The Classical Canonical Form deduced from the Rational From 58
2．The Auxiliary Unit Matrix 62
3．The Canonical Form of Jacobi 64
4．The Classical Canonical Form deduced from that of Jacobi 66
5．Uniqueness of the Classical Form:Elementary Divisors 69
6．Scalar Functions of a Square Matrix．Convergence 73
7．The Canonical Form of a Scalar Matrix Function 75
8．Matrix Determinants:Sylvester＇s Interpolation Formula 76
9．The Segre Characteristic and the Rank of Matrix Powers 79
10．Historical Note 80
CHAPTER Ⅶ CONGRUENT AND CONJUNCTIVE TRANSFORMATIONS:QUANRATIC AND HERMITIAN FORMS 82
1．The Congruent Reduction of a Conic 82
2．The Symmetrical Bilinear Form 83
3．Genralized Quadratic Forms and Congruent Transformations 84
4．The Rational Reduction of Quadratic and Hermitian Forms 85
5．The Rank of a Quadratic or Hermitian Form 86
6．The Congruent Reduction of a Skew Bilinear Form 87
7．Definite and Indefinite Forms．Sylvester＇s Law of Inertia 89
8．Determinantal Theorems concerning Rank and Index 90
9．Congruent Reduction of a General Matrix to Canonical Form 94
10．The Orthogonalizing Process of Schmidt 95
11．Observations on Schmidt＇s Theorem 96
12．Historical Note 98
CHAPTER Ⅷ CANONICAL RECUCTION BY UNITARY AND ORTHOGONAL TRANSFORMATIONS 100
1．The Latent Roots of Hermitian and Real Symmetric Matrices 100
2．The Concept of Rotation Generalized 102
3．The Canonical Reduction of Pairs of Forms or Matrices 106
4．Historical Note 111
CHAPTER Ⅸ THE CANONICAL REDUCTION OF PENCILS OF MATRICES 113
1．Singular and Non-Singular Pencils 114
2．Equivalent Canonical Reduction in the Non-Singular Case 115
3．The Invariant Factors of a Matrix Pencil 116
4．Invariance under Change of Basis 117
5．The Dependence of Vectors with Binary Linear Elements．Minimal Indices 119
6．The Canonical Minimal Submatrix,and the Vector of Apolarity 121
7．The Rational Reduction of a Singular Pencil 125
8．The Invariants of a Singular Pencil of Matrices 128
9．Application to Singular Pencils of Bilinear Forms 129
10．Quadratic and Hermitian Pencils 130
11．Weierstrass＇s Canonical Pencil of Quadratic Forms 131
12．Rational Canonical Form for Hermitian and Quadratic Pencils 133
13．Singular Hermitian and Quadratic Pencils 134
14．Reduction of a Pencil with a Basis of Transposed Matrices 135
15．Rational Canonical Form of the Foregoing Pencil 140
16．Historical Note 141
CHAPTER Ⅹ APPLICATIONS OF CANONICAL FORMS TO SOLUTION OF LINEAR MATRIX EQUATIONS．COMMUTANTS AND INVARIANTS 143
1．The Auxiliary Unit Matrices 143
2．Commutants 147
3．Scalar Function of a Matrix 149
4．Connexion between Matrix Functions and Quantum Algebra 150
5．Scalar Functions of Two Matrix Variables 151
6．Symmetric Matrices and Resolution into Factors 152
7．Invariants or Latent Forms of a Matrix 154
8．Latent Quadratic Forms 155
9．The Resolvent of a Matrix 160
10．The Adjoint Matrix and the Bordered Determinant 161
11．Orthogonal Properties of the Partial Resolvents 163
12．Application to Symmetric Matrices．Reduction by Darboux 164
13．Historical Note 166
CHAPTER Ⅺ PRACTICAL APPLICATIONS OF CANONICAL REDUCTION 167
1．The Maximum and Minimum of a Quadratic Form 167
2．Maxima and Minima of a Real Function 168
3．Conditioned Maxima and Minima of Quadratic Forms 170
4．The Vibration of a Dynamical System about Equilibrium 171
5．Matrices and Quadratic Forms in Mathematical Statistics 173
6．Sets of Linear Operational Equations with Constant Coefficients 176
7．Historical Note 178
MISCELLANEOUS EXAMPLES 180
INDEX 187