An Introduction to the Theory of Canonical Matrices

by H. W. Turnbull, A. C. Aitken

Overview

Thorough and self-contained, this penetrating study of the theory of canonical matrices presents a detailed consideration of all the theory’s principal features — from definitions and fundamental properties of matrices to the practical applications of their reduction to canonical forms.
Beginning with matrix multiplication, reciprocals, and partitioned matrices, the text proceeds to elementary transformations and bilinear and quadratic forms. A discussion of the canonical reduction of equivalent matrices follows, including treatments of general linear transformations, equivalent matrices in a field, the H. C. F. process for polynomials, and Smith’s canonical form for equivalent matrices. Subsequent chapters treat subgroups of the group of equivalent transformations and collineatory groups, discussing both rational and classical canonical forms for the latter.
Examinations of the quadratic and Hermitian forms of congruent and conjunctive transformative serve as preparation for the methods of canonical reduction explored in the final chapters. These methods include canonical reduction by unitary and orthogonal transformation, canonical reduction of pencils of matrices using invariant factors, the theory of commutants, and the application of canonical forms to the solution of linear matrix equations. The final chapter demonstrates the application of canonical reductions to the determination of the maxima and minima of a real function, solving the equations of the vibrations of a dynamical system, and evaluating integrals occurring in statistics.

Product Details

ISBN-13:	9780486153469
Publisher:	Dover Publications
Publication date:	03/05/2014
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	224
File size:	29 MB
Note:	This product may take a few minutes to download.

Table of Contents

Chapter I	Definitions and Fundamental Properties of Matrices
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
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
1	General Linear Transformation	19
2	Equivalent Matrices in a Field	19
3	The Equivalence of Matrices with Integer Elements	21
4	Polynomials with Matrix Coefficients: [lambda]-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 [lambda]-Matrix	25
8	Equivalent [lambda]-Matrices	26
9	Observations on the Theorems	27
10	The Singular Case of n Linear Equations in n Variables	29
11	Historical Note	31
Chapter IV	Subgroups of the Group of Equivalent Transformations
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
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
1	The Classical Canonical Form deduced from the Rational Form	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: Quadratic and Hermitian Forms
1	The Congruent Reduction of a Conic	82
2	The Symmetrical Bilinear Form	83
3	Generalized 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 Reduction by Unitary and Orthogonal Transformations
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
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
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
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
	Appendix	180
	Miscellaneous Examples	189
	Index	195