LAX, P. - Linear Algebra, Notas de estudo de Matemática

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LINEAR ALGEBRA

PETER D. LAX.

New York University

A Wiley-Interscience Publication JOHN WILEY & SONS, INC.

-.NewYork. Chichester. Brisbane. Toronto. Singapore. Weinheim

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CONTENTS

Preface

1. Fundamentals

Linear Space, Isomorphism, Subspace, 2 Linear Dependence, 3 Basis, Dimension, 3 Quotient Space, 5

2. Duality Linear Functions, 8 Annihilator, II Codimension, 11 Quadrature Formula, 12
3. Linear Mappings

Domain and Target Space, 14 Nullspace and Range, 15 Fundamental Theorem, 15 Underdetermined Linear Systems, 16 Interpolation, 17 Difference Equations, 17 AI2:ebra of Linear Mappings,proJtl,UVu,-" _ _ ~ 18

4. Matrices

Rows and Columns, 26 ~.. .:.. "p'''nltinlication, 27

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8. Spectral Theory of Selfadjoint Mappings

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CONTENTS

13. The Duality Theorem Farkas-Minkowski Theorem, Duality Theorem, 173 Economics Interpretation, 175 Minmax Theorem, 177

14. Normed Linear Spaces Norm, 180 I" Norms, 18I Equivalence of Norms, 183 DualNorm, 185 , Distance from Subspace, 187 Normed Quotient Space, 188 Complex Normed Spaces, 189

15. Linear Mappings between Normed Spaces

Norm of a Mapping, 191 Norm of Transpose, 192 Normed Algebra of Maps, 193 Invertible Maps, 193

16. Positive Matrices

Perron's Theorem, 196 Stochastic Matrices, 199 Frobenius' Theorem, 202

17. How to Solve Systems of Linear Equations

History, 205 Condition Number, 206 Iterative Methods, 207 Steepest Descent, 208 Chebychev Iteration, 211 Three-term Chebychev Iteration, 214 Optimal Three-term Recursion Relation, 215 Rate of Convergence, 219

Appendix 1. Special Determinants ' 221

Appendix 2. Pfaff'sTheorem 224

Appendix 3. Symplectic Matrices 227

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CONTENTS

Appendix 4. Tensor Product

Appendix 5. Lattices

Appendix 6. Fast Matrix Multiplication

Appendix 7. Gershgorin's Theorem

Appendix 8. The Multiplicity of Eigenvalues

Bibliography

Index

List of Series Titles

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xii PREFACE

particularly exploit quotient spaces as a counting device. This dry material is enlivened by some nontrivial applications to quadrature, to interpolation by polynomials and to solving the Dirichlet problem for the discretized Laplace equation. In Chapter 5 determinants are motivated geometrically as signed volumes of ordered simplices. The basic algebraic properties of determinants follow immediately. Chapter 6 is devoted to the spectral theory of arbitrary square matrices with complex entries. The completeness of eigenvectors and generalized eigenvectors is proved without the characteristic equation, relying only on the divisibility theory of the algebra of polynomials. In the same spirit we show that two matrices A and Bare sim.ilar if and only if (A - kI)/J/ and (B - kIt' have nullspaces of the same dimension for all corrfplex k an.d all positive integer m.. The proof of this proposition leads to the Jordan canonical form. Euclidean structure appears for the first time in Chapter' 7. It is used in Chapter 8 to derive the spectral theory of selfadjoint matric~s. We present two proofs, one based on the sp~ctral theory of general matrices, the other using the variational characterization of eigenvectors and eigenvalues. Fischer's minmax theorem is explained. Chapter 9 deals with the calculus of vector and matrix valued functions of a single variable, an important topic not usually discussed in the undergraduate curriculum. The most important result is the continuous and differentiable character of eigenvalues and normalized eigenvectors of differentiable matrix functions, provided that appropriate nondegeneracy conditions are satisfied. The fascinating phenomenon of "avoided crossings" is briefly described and explained. The first nine chapters, or certainly the first eight, constitute the core of linear algebra. The next eight chapters deal with special topics, to be taken up depending on the interest of the instructor and of the students. We shall comment on them very briefly. Chap~er 10 is a symphony of inequalities about matrices, their eigenvalues, and their determinants. Many of the proofs make use of calculus. I included Chapter 11 to make up for the unfortunate disappearance of mechanics from the curriculum and to show how matrices give an elegant description of motion in space. Anguluar velocity of a rigid body and divergence and curl of a vector field all appear naturally. The monotonic dependence of eigenvalues of symmetric matrices is used to show that the natural frequencies of a vibrating system increase if the system is stiffened and the masses are decreased.

Chapters 12, 13, and 14 are linked together by the notion of convexity. In Chapter 12 we present the descriptions of convex sets in terms of gauge functions and support functions. The workhorse of the subject, the hyperplane separation theorem, is proved by means of the Hahn-Banach procedure. Caratheodory's theorem on extreme points is proved and used to derive the

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and for trying out parts of it on their classes. I am grateful to Connie Engle and Janice Want for their expert typing. I have learned a great deal from Richard Bellman's'outstanding book, Intro- duction to Matrix Analysis; its influence on the present volume is considerable. For this reason and to mark a friendship that began in 1945 and lasted until his death in 1984, I dedicate this book to his memory. L

PETER D. LAX

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Isomorphic linear spaces are indistinguishable by means of operations avail- able in linear spaces. Two linear spaces that are presented in very different ways can be, as we shall see, isomorphic.

Examples of Linear Spaces. (i) Set of all row vectors: (a I'... , all)' ai in K; addition. multiplication defined componentwise. This space is denoted as K". (ii) Set of all real valued functions f(x) defined on the real line, K = IR. (iii) Set of all functions with values in K, defined on an arbitrary set S. (iv) Set of all polynomials of degree less than 11with coefficients in K.

EXERCISEJ. Show that (i) and (iv) are isomorphic. , EXERCISE~.Show that if S has n elements, (i) is the same as (iii). I EXERCISE5. Show that when K = IR, (iv) is isomorphic with (iii) when S con- sists of n distinct points of IR.

Definition. A subset Y of a linear space X is called a subspace if sums and scalar multiples of elements of Y belong to Y.

Examples of Subspaces. (a) X as in Example (i). Y the set of vectors (0, a2.... "all-I' 0) whose first and last component is zero. (b) X as in Example (ii), Y the set of all periodic functions with period 7T. (c) X as in Example (iii). Y the set of constant functions on S. (d) X as in Example (iv), Y the set of all even polynomials.

Definition. The sum of two subsets Y and Z of a linear space X, denoted as Y + Z, is'the set of all vectors of form y + Z" Y in Y, :: in Z.

EXERCISE6. Prove that Y + Z is a linear subspace of X if Y and Z are.

Definition. The intersection of two subsets Y and Z of a linear space X. denoted as Y n Z, consists of all vectors x that belong to both Y and Z.

EXERCISE7. Prove that if Y and Z are linear subspaces of X, so is Y n z.

EXERCISEX. Show that the set {O}consisting of the zero element of a linear space X is a linear subspace of X. It is called the trivial subspace.

Definition. A linear combination of j vectors xi,... , Xi of a linear space is a vector of the form

EXERCISEY. Show that the set of all linear combinations of XI'... , Xi is the smallest linear subspace of X containing x I'... 'Xi' This is called the subspace spanned by XI'.... xi'

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