p ; 5̂ Introduction to Matrix Analysis

±j Second Edition ^-^

Richard Bellman

Society for Industrial and Applied Mathetnatics

Philadelphia

Contents

Foreword xvii

Preface to Second Edition xix

Preface xxi

Chapter 1. Maximization, Minimization, and Motivation 1 Introduction 1 Maximization of Functions of One Variable 1 Maximization of Functions of Two Variables 2 Algebraic Approach 3 Analytic Approach 4 Analytic Approach-II 6 A Simplifying Transformation . 7 Another Necessary and Sufficient Condition 8 Definite and Indefinite Forms 8 Geometrie Approach 9 Discussion 10

Chapter 2. Vectors and Matrices 12 Introduction 12 Vectors 12 Vector Addition 13 Scalar Multiplication 14 The Inner Product of Two Vectors 14 Orthogonality 15 Matrices 16 Matrix Multiplication-Vector by Matrix 17 Matrix Multiplication-Matrix by Matrix 18 Noncommutativity 20 Associativity 20 Invariant Vectors 21 Quadratic Forms as Inner Products 22 The Transpose Matrix 23 Symmetrie Matrices 23 Hermitian Matrices 24 Invariance of Distance-Orthogonal Matrices 25 Unitary Matrices 25

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viii Contents

Chapter 3. Diagonalization and Canonical Forms for Symmetrie Matrices 32 Recapitulation 32 The Solution of Linear Homogeneous Equations 32 Characteristic Roots and Vectors 34 Two Fundamental Properties of Symmetrie Matrices 35 Reduction to Diagonal Form-Distinct Characteristic Roots 37 Reduction of Quadratäc Forms to Canonical Form 39 Positive Definite Quadratic Forms and Matrices 40

Chapter 4. Reduction of General Symmetrie Matrices to Diagonal Form 44 Introduction 44 Linear Dependence 44 Gram-Schmidt Orthogonalization 44 On the Positivity of the D t 47 Anldentity 49 The Diagonalization of General Symmetrie Matrices-Two-dimensional 50 N-dimensional Case 51 A Necessary and Sufficient Condition for Positive Definiteness 54 Characteristic Vectors Associated with Multiple Characteristic Roots 54 The Cayley-Hamilton Theorem for Symmetrie Matrices 55 Simultaneous Reduction to Diagonal Form 56 Simultaneous Reduction to Sum of Squares 58 Hermitian Matrices 59 The Original Maximization Problem 60 Perturbation Theory-1 60 Perturbation Theory-II 61

Chapter 5. Constrained Maxima 73 Introduction 73 Determinantal Criteria for Positive Definiteness 73 Representation as Sum of Squares 75 Constrained Variation and Finsler's Theorem 76 The Case k = 1 78 A Minimization Problem 81 General Values of k 82 Rectangular Arrays 82

Contents ix

Composite Matrices 83 The Result for General k 85

Chapter 6. Functions of Matrices 90 Introduction 90 Functions of Symmetrie Matrices 90 The Inverse Matrix 91 Uniqueness of Inverse 91 Square Roots 93 Parametric Representation 94 A Result of I. Schur 95 The Fundamental Scalar Functions 95 The Infinite Integral £ V u , w d x 97 An Analogue for Hermitian Matrices 99 Relation between J(H) and \H\ 99

Chapter 7. Variational Description of Characteristic Roots 112 Introduction 112 The Rayleigh Quotient 112 Variational Description of Characteristic Roots 113 Discussion 114 Geometrie Preliminary 114 The Courant-Fischer min-max Theorem 115 Monotone Behavior of Xk(A) 117 A Sturmian Separation Theorem 117 A Necessary and Suffident Condition that A Be Positive Definite 118 The Poincare Separation Theorem 118 A Representa'"on Theorem 119 Approximate Techniques 120

Chapter 8. Inequalities 126 Introduction 126 The Cauchy-Schwarz Inequality 126 Integral Version 126 Holder Inequality 127 Concavity of | A | 128 A Useful Inequality 129 Hadamard's Inequality 130 Concavity of A,NXm• • •'kk 130 Additive Inequalities from Multiplicative 131 An Alternate Route 132 A Simpler Expression for Ä.wA,N.1"

,At 133

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Arithmetic-Geometric Mean Inequality 134 Multiplicative Inequalities from Additive 135

Chapter 9. Dynamic Programming 144 Introduction 144 A Problem of Minimum Deviation 144 Functional Equations 145 Recurrence Relations 146 A More Complicated Example 146 Sturm-Liouville Problems 147 Functional Equations 148 Jacobi Matrices 150 Analytic Continuation 151 Nonsymmetric Matrices 151 Complex A 152 Slightly Intertwined Systems 153 Simplifications-1 154 Simplifications-II 154 The Equation Ax = y 155 Quadratic Deviation 156 A Result of Stieltjes 157

Chapter 10. Matrices and Differential Equations 163 Motivation 163 Vector-matrix Notation 164 Norms of Vectors and Matrices 165 Infinite Series of Vectors and Matrices 167 Existence and Uniqueness of Solutions of Linear Systems 167 The Matrix Exponential 169 Functional Equations-1 170 Functional Equations-II 171 Functional Equations-III 171 Nonsingularity of Solution 171 Solution of Inhomogeneous Equation-Constant Coefficients 173 Inhomogeneous Equation-Variable Coefficients 173 Inhomogeneous Equation-Adjoint Equation 174 Perturbation Theory 174 Non-negativity of Solution 176 Polya's Functional Equation 177 The Equation dX/dt = AX+ XB 179

Contents XI

Chapter 11. Explicit Solutions and Canonical Forms .190 Introduction 190 Euler'sMethod.. 190 Construction of a Solution 191 Nonsingularity of C 191 SecondMethod 192 The Vandermonde Determinant 193 Explicit Solution of Linear Differential Equations— Diagonal Matrices 194 Diagonalization of a Matrix 194 Connection between Approaches 195 Multiple Characteristic Roots 197 Jordan Canonical Form 198 Multiple Characteristic Roots 199 Semidiagonal or Triangulär Form-Schur's Theorem 202 Normal Matrices 204 An Approximation Theorem 205 Another Approximation Theorem 206 The Cayley-Hamilton Theorem 207 Alternate Proof of Cayley-Hamilton Theorem 207 Linear Equations with Periodic Coefficients 208 A Nonsingular Matrix Is an Exponential 209 An Alternate Proof 211 Some Interesting Transformations 212 Biorthogonality 213 The Laplace Transform 215 An Example 216 Discussion 217 Matrix Case 218

Chapter 12. Symmetrie Function, Kronecker Products and Circulants 231 Introduction 231 Powers of Characteristic Roots 231 Polynomials and Characteristic Equations 233 Symmetrie Functions 234 Kronecker Products 235 Algebra of Kronecker Products 236 Kronecker Powers-1 236 Kronecker Powers—II 236 Kronecker Powers-III 237 Kronecker Logarithm 237 Kronecker Sum-1 238

xii Contents

Kronecker Sum-II 238 The Equation AX+ XB - Q The Lyapunov Equation 239 An Alternate Route 240 Circulants 242

Chapter 13. Stability Theory 249 Introduction 249 A Necessary and Sufficient Condition for Stability 249 Stability Matrices 251 A Method of Lyapunov 251 Mean-square Deviation 252 Effective Tests for Stability 253 A Necessary and Sufficient Condition for Stability Matrices 254 Differential Equations and Characteristic Values 255 Effective Tests for Stability Matrices 256

Chapter 14. Markoff Matrices and Probability Theory 263 Introduction 263 A Simple Stochastic Process 263 Markoff Matrices and Probability Vectors 265 Analytic Formulation of Discrete Markoff Processes 265 Asymptotic Behavior 266 First Proof 266 Second Proof of Independence of Initial State 268 Some Properties of Positive Markoff Matrices 269 Second Proof of Limiting Behavior 270 General Markoff Matrices 271 A Continuous Stochastic Process 272 Proof of Probabilistic Behavior 274 Generalized Probabilities-Unitary Transformations 274 Generalized Probabilities—Matrix Transformations 275

Chapter 15. Stochastic Matrices 281 Introduction 281 Limiting Behavior of Physical Systems 281 Expected Values 282 Expected Values of Squares 283

Chapter 16. Positive Matrices, Perron's Theorem, and Mathematical Economics 286 Introduction 286 Some Simple Growth Processes 286

Contents xiii

Notation 287 The Theorem of Perron 288 Proof of Theorem 1 288 First Proof That X(A) Is Simple 290 Second Proof of the Simplicity of A,(A) 290 Proof of the Minimum Property of \(A) 291 An Equivalent Definition of X(A)... 292 A Limit Theorem 292 Steady-state Growth 292 Continuous Growth Processes 293 Analogue of Perron Theorem 293 Nuclear Fission 294 Mathematical Economics 294 Minkowski-Leontieff Matrices 298 Positivity of | J - A| 298 Strengthening of Theorem 6 299 Linear Programming 299 The Theory of Games 300 A Markovian Decision Process 301 An Economic Model 302

Chapter 17. Control Processes 316 Introduction 316 Maintenance of Unstable Equilibrium 316 Discussion 317 The Euler Equation for A = 0 318 Discussion 319 Two-point Boundary-value Problem 320 Nonsingularity of X^(T) 320 Analytic Form of Solution 321 Alternate Forms and Asymptotic Behavior 322 Representation of the Square Root of ß 322 Riccati Differential Equation for R 323 Instability of Euler Equation 323 Proof that J(x) Attains an Absolute Minimum 324 Dynamic Programming 325 Dynamic Programming Formalism 325 Riccati Differential Equation 327 Discussion 328 More General Control Processes 328 Discrete Deterministic Control Processes 329 Discrete Stochastic Control Processes 331

xiv Contents

Potential Equation 331 Discretized Criterion Function 332

Chapter 18. Invariant Imbedding 338 Introduction 338 Terminal Condition as a Function of a 338 Discussion 340 Linear Transport Process 340 Classical Imbedding 341 Invariant Imbedding 342 Interpretation 344 Conservation Relation 345 Non-negativity of R and T 346 Discussion 347 Internal Fluxes 347 Absorption Processes 348

Chapter 19. Numerical Inversion of the Laplace Transform and Tychonov Regularization 353 Introduction 353 The Heat Equation 354 The Renewal Equation 355 Differential-difference Equations 355 Discussion 355 Instability 356 Quadrature 356 Gaussian Quadrature 357 Approximating System of Linear Equations 359 A Device of Jacobi 359 Discussion 361 Tychonov Regularization 362 The Equation (p(x) = X,(x - c, x- c) 363 öbtaining an Initial Approximation 364 Self-consistent (p(x) 365 Nonlinear Equations 365 Error Analysis 366

Appendix A. Linear Equations and Rank 371 Introduction 371 Determinants 371 A Property of Cofactors ...372 Cramer's Rule 372 Homogeneous Systems 372

Contents xv

Rank 376 Rank of a Quadratic Form 377 Law of Inertia 377 Signature 377

Appendix B. The Quadratic Form of Seiberg 379

Appendix C. A Method of Hermite 383

Appendix D. Moments and Quadratic Forms 385 Introduction 385 A Device of Stieltjes 385 A Technique of E. Fischer , 386 Representation as Moments 387 A Result of Herglotz 388

Indexes 391