The Theory of Matrices, Volume 1

Front Cover
PREFACE
PUBLISHERS'
PREFACE
CONTENTS
CHAPTER I: MATRICES AND OPERATIONS ON MATRICES
I. Matrices. Basic Notation
2. Addition and Multiplication of Rectangular Matrices
3. Square Matrices
4. Compound Matrices. Minors of the Inverse Matrix
CHAPTER II: THE ALGORITHM OF GAUSS AND SOME OF ITS APPLICATIONS
1. Gauss's Elimination Method
2. Mechanical Interpretation of Gauss's Algorithm
3. Sylvester's Determinant Identity
4. The Decomposition of a Square Matrix into Triangular Factors
5. The Partition of a Matrix into Blocks. The Technique of Operating with Partitioned Matrices. The Generalized Algorithm of Gauss
CHAPTER III: LINEAR OPERATORS IN AN n-DIMENSIONAL VECTOR SPACE
I. Vector Spaces
2. A Linear Operator Mapping an n-Dimensional Space into an m-Dimensional Space
3. Addition and Multiplication of Linear Operators
4. Transformation of Coordinates
5. Equivalent Matrices. The Rank of an Operator. Sylvester's Inequality
6. Linear Operators Mapping an n-Dimensional Space into Itself
7. Characteristic Values and Characteristic Vectors of a Linear Operator
8. Linear Operators of Simple Structure
CHAPTER IV: THE CHARACTERISTIC POLYNOMIAL AND THE MINIMAL POLYNOMIAL OF A MATRIX
I. Addition and Multiplication of Matrix Polynomials
2. Right and Left Division of Matrix Polynomials
3. The Generalized Bezout Theorem
4. The Characteristic Polynomial of a Matrix. The Adjoint Matrix
5. The Method of Faddeev for the Simultaneous Computation of the Coefficients of the Characteristic Polynomial and of the Adjoint Matrix
6. The Minimal Polynomial of a Matrix
CHAPTER V: FUNCTIONS OF MATRICES
I. Definition of a Function of a Matrix
2. The Lagrange-Sylvester Interpolation Polynomial
3. Other Forms of the Definition of f(A). The Components of the Matrix A
4. Representation of Functions of Matrices by means of Series
5. Application of a Function of a Matrix to the Integration of a System of Linear Differential Equations with Constant Coefficients
6. Stability of Motion in the Case of a Linear System
CHAPTER VI: EQUIVALENT TRANSFORMATIONS OF POLYNOMIAL MATRICES. ANALYTIC THEORY OF ELEMENTARY DIVISORS
1. Elementary Transformations of a Polynomial Matrix
2. Canonical Form of a ?-Matrix
3. Invariant Polynomials and Elementary Divisors of a Polynomial Matrix
4. Equivalence of Linear Binomials
5. A Criterion for Similarity of Matrices
6. The Normal Forms of a Matrix
7. The Elementary Divisors of the Matrix f(A.)
8. A General Method of Constructing the Transforming Matrix
9. Another Method of Constructing a Transforming Matrix
CHAPTER VII: THE STRUCTURE OF A LINEAR OPERATOR IN AN n-DIMENSIONAL SPACE (Geometrical Theory of Elementary Divisors)
1. The Minimal Polynomial of a Vector and a Space (with Respect to a Given Linear Operator)
2. Decomposition into Invariant Subspaces with Co-Prime Minimal Polynomials
3. Congruence. Factor Space
4. Decomposition of a Space into Cyclic Invariant Subspaces
5. The Normal Form of a Matrix
6. Invariant Polynomials, Elementary Divisors
7. The Jordan Normal Form of a Matrix
8. K.rylov's Method of Transforming the Secular Equation
CHAPTER VIII: MATRIX EQUATIONS
1. The Equation AX= XB
2. The Special Case .4 = B. Commuting Matrices
3. The Equation AX - XB = C
4. The Scalar Equation f(X) = 0
5. Matrix Polynomial Equations
6. The Extraction of m-th Roots of a Non-Singular Matrix
7. The Extraction of m-th Roots of a Singular Matrix
8. The Logarithm of a Matrix
CHAPTER IX: LINEAR OPERATORS IN A UNITARY SPACE
1. General Considerations
2. Metrization of a Space
3. Gram's Criterion for Linear Dependence of Vectors
4. Orthogonal Projection
5. The Geometrical Meaning of the Gramian and Some Inequalities
6. Orthogonalization of a Sequence of Vectors
7. Orthonormal Bases
8. The Adjoint Operator
9. Normal Operators in a Unitary Space
10. The Spectra of Normal, Hermitian, and Unitary Operators
11. Positive-Semidefinite and Positive-Definite Hermitian Operators
12. Polar Decomposition of a Linear Operator in a Unitary Space. Cayley's Formulas
13. Linear Operators in a Euclidean Space
14. Polar Decomposition of an Operator and the Cayley Formulas in a Euclidean Space
15. Commuting Normal Operators
CHAPTER X: QUADRATIC AND HERMITIAN FORMS
I. Transformation of the Variables in a Quadratic Form
2. Reduction of a Quadratic Form to a Sum of Squares. The Law of Inertia
3. The Methods of Lagrange and Jacobi of Reducing a Quadratic Form to a Sum of Squares
4. Positive Quadratic Forms
5. Reduction of a Quadratic Form to Principal Axes
6. Pencils of Quadratic Forms
7. Extremal Properties of the Characteristic Values of a Regular Pencil of Forms
8. Small Oscillations of a System with n Degrees of Freedom
9. Hermitian Forms
10. Hankel Forms
BIBLIOGRAPHY
INDEX
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