Book:L. Mirsky/An Introduction to Linear Algebra

Subject Matter

 * Linear Algebra

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


 * 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


 * 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


 * 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


 * 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


 * 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


 * 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


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


 * 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


 * 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


 * 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


 * 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


 * 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


 * BIBLIOGRAPHY


 * INDEX