Book:L. Mirsky/An Introduction to Linear Algebra
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L. Mirsky: An Introduction to Linear Algebra
Published $\text {1955}$, Oxford at the Clarendon Press
Subject Matter
Contents
- Preface
- PART $\text {I}$: DETERMINANTS, VECTORS, MATRICES, AND LINEAR EQUATIONS
- $\text {I}$. 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
- $\text {I}$. DETERMINANTS
- $\text {II}$. 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
- $\text {II}$. VECTOR SPACES AND LINEAR MANIFOLDS
- $\text {III}$. 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
- $\text {III}$. THE ALGEBRA OF MATRICES
- $\text {IV}$. 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
- $\text {IV}$. LINEAR OPERATORS
- $\text {V}$. 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
- $\text {V}$. SYSTEMS OF LINEAR EQUATIONS AND RANK OF MATRICES
- $\text {VI}$. 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
- $\text {VI}$. ELEMENTARY OPERATIONS AND THE CONCEPT OF EQUIVALENCE
- PART $\text {II}$: FURTHER DEVELOPMENT OF MATRIX THEORY
- $\text {VII}$. 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
- $\text {VII}$. THE CHARACTERISTIC EQUATION
- $\text {VIII}$. ORTHOGONAL AND UNITARY MATRICES
- 8.1. Orthogonal matrices
- 8.2. Unitary matrices
- 8.3. Rotations in the plane
- 8.4. Rotations in space
- $\text {VIII}$. ORTHOGONAL AND UNITARY MATRICES
- $\text {IX}$. 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
- $\text {IX}$. GROUPS
- $\text {X}$. 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
- $\text {X}$. CANONICAL FORMS
- $\text {XI}$. 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
- $\text {XI}$. MATRIX ANALYSIS
- PART $\text {III}$: QUADRATIC FORMS
- $\text {XII}$. 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
- $\text {XII}$. BILINEAR, QUADRATIC, AND HERMITIAN FORMS
- $\text {XIII}$. 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
- $\text {XIII}$. DEFINITE AND INDEFINITE FORMS
- BIBLIOGRAPHY
- INDEX
Source work progress
- 1955: L. Mirsky: An Introduction to Linear Algebra ... (next): Chapter $\text I$: Determinants