# Book:L. Mirsky/An Introduction to Linear Algebra

## L. Mirsky: *An Introduction to Linear Algebra*

Published $1955$, **Oxford at the Clarendon Press**.

### Subject Matter

### Contents

- Preface

- PART I: DETERMINANTS, VECTORS, MATRICES, AND LINEAR EQUATIONS

- I. DETERMINANTS
- 1.1. Arrangements and the $\epsilon$-symbol
- 1.2. Elementary properties of determinants
- 1.3. Multiplication of determinants
- 1.4. Expansion theorems
- 1.5. Jacobi's theorem
- 1.6. Two special theorems on linear equations

- I. DETERMINANTS

- II. VECTOR SPACES AND LINEAR MANIFOLDS
- 2.1. The algebra of vectors
- 2.2. Linear manifolds
- 2.3. Linear dependence and bases
- 2.4. Vector representation of linear manifolds
- 2.5. Inner products and orthonormal bases

- II. VECTOR SPACES AND LINEAR MANIFOLDS

- III. THE ALGEBRA OF MATRICES
- 3.1. Elementary algebra
- 3.2. Preliminary notions concerning matrices
- 3.3. Addition and multiplication of matrices
- 3.4. Application of matrix technique to linear substitutions
- 3.5. Adjugate matrices
- 3.6. Inverse matrices
- 3.7. Rational functions of a square matrix
- 3.8. Partitioned matrices

- III. THE ALGEBRA OF MATRICES

- IV. LINEAR OPERATORS
- 4.1. Change of basis in a linear manifold
- 4.2. Linear operators and their representations
- 4.3. Isomorphisms and automorphisms of linear manifolds
- 4.4. Further instances of linear operators

- IV. LINEAR OPERATORS

- V. SYSTEMS OF LINEAR EQUATIONS AND RANK OF MATRICES
- 5.1. Preliminary results
- 5.2. The rank theorem
- 5.3. The general theory of linear equations
- 5.4. Systems of homogeneous linear equations
- 5.5. Miscellaneous applications
- 5.6. Further theorems on rank of matrices

- V. SYSTEMS OF LINEAR EQUATIONS AND RANK OF MATRICES

- VI. ELEMENTARY OPERATIONS AND THE CONCEPT OF EQUIVALENCE
- 6.1. $E$-operations and $E$-matrices
- 6.2. Equivalent matrices
- 6.3. Applications of the preceding theory
- 6.4. Congruence transformations
- 6.5. The general concept of equivalence
- 6.6. Axiomatic characterization of determinants

- VI. ELEMENTARY OPERATIONS AND THE CONCEPT OF EQUIVALENCE

- PART II: FURTHER DEVELOPMENT OF MATRIX THEORY

- VII. THE CHARACTERISTIC EQUATION
- 7.1. The coefficients of the characteristic polynomial
- 7.2. Characteristic polynomials and similarity transformations
- 7.3. Characteristic roots of rational functions of matrices
- 7.4. The minimum polynomial and the theorem of Cayley and Hamilton
- 7.5. Estimates of characteristic roots
- 7.6. Characteristic vectors

- VII. THE CHARACTERISTIC EQUATION

- VIII. ORTHOGONAL AND UNITARY MATRICES
- 8.1. Orthogonal matrices
- 8.2. Unitary matrices
- 8.3. Rotations in the plane
- 8.4. Rotations in space

- VIII. ORTHOGONAL AND UNITARY MATRICES

- IX. GROUPS
- 9.1. The axioms of group theory
- 9.2. Matrix groups and operator groups
- 9.3. Representation of groups by matrices
- 9.4. Groups of singular matrices
- 9.5. Invariant spaces and groups of linear transformations

- IX. GROUPS

- X. CANONICAL FORMS
- 10.1. The idea of a canonical form
- 10.2. Diagonal canonical forms under the similarity group
- 10.3. Diagonal canonical forms under the orthogonal similarity group and the unitary similarity group
- 10.4. Triangular canonical forms
- 10.5. An intermediate canonical form
- 10.6. Simultaneous similarity transformations

- X. CANONICAL FORMS

- XI. MATRIX ANALYSIS
- 11.1. Convergent matrix sequences
- 11.2. Power series and matrix functions
- 11.3. The relation between matrix functions and matrix polynomials
- 11.4. Systems of linear differential equations

- XI. MATRIX ANALYSIS

- PART III: QUADRATIC FORMS

- XII. BILINEAR, QUADRATIC, AND HERMITIAN FORMS
- 12.1. Operators and forms of the bilinear and quadratic types
- 12.2. Orthogonal reduction to diagonal form
- 12.3. General reduction to diagonal form
- 12.4. The problem of equivalence. Rank and signature
- 12.5. Classification of quadrics
- 12.6. Hermitian forms

- XII. BILINEAR, QUADRATIC, AND HERMITIAN FORMS

- XIII. DEFINITE AND INDEFINITE FORMS
- 13.1. The value classes
- 13.2. Transformations of positive definite forms
- 13.3. Determinantal criteria
- 13.4. Simultaneous reduction of two quadratic forms
- 13.5. The inequalities of Hadamard, Minkowski, Fischer, and Oppenheim

- XIII. DEFINITE AND INDEFINITE FORMS

- BIBLIOGRAPHY

- INDEX