Linear Algebra, Theory And Applications, 2012a

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6 CONTENTS 13 Self Adjoint Operators 307 13.1 Simultaneous Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 13.2 Schur’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 13.3 Spectral Theory Of Self Adjoint Operators . . . . . . . . . . . . . . . . . . . . 312 13.4 Positive And Negative Linear Transformations . . . . . . . . . . . . . . . . . 317 13.5 Fractional Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 13.6 Polar Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 13.7 An Application To Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 13.8 The Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 327 13.9 Approximation In The Frobenius Norm . . . . . . . . . . . . . . . . . . . . . 329 13.10Least Squares And Singular Value Decomposition . . . . . . . . . . . . . . . . 331 13.11The Moore Penrose Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 13.12Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 14 Norms For Finite Dimensional Vector Spaces 337 14.1 The p Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 14.2 The Condition Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 14.3 The Spectral Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 14.4 Series And Sequences Of Linear Operators . . . . . . . . . . . . . . . . . . . . 350 14.5 Iterative Methods For Linear Systems . . . . . . . . . . . . . . . . . . . . . . 354 14.6 Theory Of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 14.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 15 Numerical Methods For Finding Eigenvalues 371 15.1 The Power Method For Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . 371 15.1.1 The Shifted Inverse Power Method . . . . . . . . . . . . . . . . . . . . 375 15.1.2 The Explicit Description Of The Method . . . . . . . . . . . . . . . . 376 15.1.3 Complex Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 15.1.4 Rayleigh Quotients And Estimates for Eigenvalues . . . . . . . . . . . 383 15.2 The QR Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 15.2.1 Basic Properties And Definition . . . . . . . . . . . . . . . . . . . . . 386 15.2.2 The Case Of Real Eigenvalues . . . . . . . . . . . . . . . . . . . . . . 390 15.2.3 The QR Algorithm In The General Case . . . . . . . . . . . . . . . . . 394 15.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 A Positive Matrices 403 B Functions Of Matrices 411 C Applications To Differential Equations 417 C.1 Theory Of Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . 417 C.2 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 C.3 Local Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 C.4 First Order Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 C.5 Geometric Theory Of Autonomous Systems . . . . . . . . . . . . . . . . . . . 428 C.6 General Geometric Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 C.7 The Stable Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 D Compactness And Completeness 439 D.0.1 The Nested Interval Lemma . . . . . . . . . . . . . . . . . . . . . . . . 439 D.0.2 Convergent Sequences, Sequential Compactness . . . . . . . . . . . . . 440
CONTENTS 7 E The Fundamental Theorem Of Algebra 443 F Fields And Field Extensions 445 F.1 The Symmetric Polynomial Theorem . . . . . . . . . . . . . . . . . . . . . . . 445 F.2 The Fundamental Theorem Of Algebra . . . . . . . . . . . . . . . . . . . . . . 447 F.3 Transcendental Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 F.4 More On Algebraic Field Extensions . . . . . . . . . . . . . . . . . . . . . . . 459 F.5 The Galois Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 F.6 Normal Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 F.7 Normal Extensions And Normal Subgroups . . . . . . . . . . . . . . . . . . . 470 F.8 Conditions For Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 F.9 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 F.10 Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 F.11 Solvability By Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 G Answers To Selected Exercises 487 G.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 G.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 G.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 G.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 G.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 G.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 G.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 G.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 G.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 G.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 G.11Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 G.12Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 G.13Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 G.14Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 G.15Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 G.16Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 G.17Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 G.18Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 G.19Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 G.20Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 G.21Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 G.22Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 G.23Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 Copyright c○ 2012,
Page 1 and 2: Linear Algebra, Theory And Applicat
Page 3 and 4: Contents 1 Preliminaries 11 1.1 Set
Page 5: CONTENTS 5 8 Vector Spaces And Fiel
Page 9 and 10: Preface This is a book on linear al
Page 11 and 12: Preliminaries 1.1 Sets And Set Nota
Page 13 and 14: 1.3. THE NUMBER LINE AND ALGEBRA OF
Page 15 and 16: 1.5. THE COMPLEX NUMBERS 15 Also fr
Page 17 and 18: 1.5. THE COMPLEX NUMBERS 17 and so
Page 19 and 20: 1.6. EXERCISES 19 Eventually, for r
Page 21 and 22: 1.8. WELL ORDERING AND ARCHIMEDEAN
Page 23 and 24: 1.9. DIVISION AND NUMBERS 23 Theore
Page 25 and 26: 1.9. DIVISION AND NUMBERS 25 Exampl
Page 27 and 28: 1.10. SYSTEMS OF EQUATIONS 27 Why i
Page 29 and 30: 1.10. SYSTEMS OF EQUATIONS 29 The a
Page 31 and 32: 1.11. EXERCISES 31 It follows x =
Page 33 and 34: 1.14. EXERCISES 33 the existence of
Page 35 and 36: 1.15. THE INNER PRODUCT IN F N 35 s
Page 37 and 38: Matrices And Linear Transformations
Page 39 and 40: 2.1. MATRICES 39 Definition 2.1.2 M
Page 41 and 42: 2.1. MATRICES 41 is it possible to
Page 43 and 44: 2.1. MATRICES 43 a3× 3 matrix as d
Page 45 and 46: 2.1. MATRICES 45 1 2 3 4 Write the
Page 47 and 48: = ∑ k ( B T ) ik ( A T ) kj 2.1.
Page 49 and 50: 2.1. MATRICES 49 As described above
Page 51 and 52: 2.2. EXERCISES 51 Write the augment
Page 53 and 54: 2.3. LINEAR TRANSFORMATIONS 53 (e)
Page 55 and 56: 2.3. LINEAR TRANSFORMATIONS 55 Proo
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2.4. SUBSPACES AND SPANS 57 Proof:
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2.4. SUBSPACES AND SPANS 59 Proof 3
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2.5. AN APPLICATION TO MATRICES 61
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2.6. MATRICES AND CALCULUS 63 2.6.1
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2.6. MATRICES AND CALCULUS 65 where
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2.6. MATRICES AND CALCULUS 67 There
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2.6. MATRICES AND CALCULUS 69 Using
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2.7. EXERCISES 71 this will suffice
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2.7. EXERCISES 73 22. If A is a lin
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2.7. EXERCISES 75 35. ↑Suppose yo
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Determinants 3.1 Basic Techniques A
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3.1. BASIC TECHNIQUES AND PROPERTIE
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3.2. EXERCISES 81 First note det (A
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3.3. THE MATHEMATICAL THEORY OF DET
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3.4. THE CAYLEY HAMILTON THEOREM 97
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3.5. BLOCK MULTIPLICATION OF MATRIC
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3.5. BLOCK MULTIPLICATION OF MATRIC
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3.6. EXERCISES 103 derivatives so i
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Row Operations 4.1 Elementary Matri
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4.1. ELEMENTARY MATRICES 107 This h
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4.1. ELEMENTARY MATRICES 109 Now co
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4.2. THE RANK OF A MATRIX 111 So ho
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4.3. THE ROW REDUCED ECHELON FORM 1
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4.3. THE ROW REDUCED ECHELON FORM 1
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4.5. FREDHOLM ALTERNATIVE 117 Proof
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4.6. EXERCISES 119 5. ↑Consider t
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4.6. EXERCISES 121 18. Suppose A is
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Some Factorizations 5.1 LU Factoriz
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5.3. SOLVING LINEAR SYSTEMS USING A
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5.5. JUSTIFICATION FOR THE MULTIPLI
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5.6. EXISTENCE FOR THE PLU FACTORIZ
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5.7. THE QR FACTORIZATION 131 so it
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5.8. EXERCISES 133 Theorem 5.7.5 Le
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Linear Programming 6.1 Simple Geome
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6.2. THE SIMPLEX TABLEAU 137 set x
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6.2. THE SIMPLEX TABLEAU 139 lution
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6.3. THE SIMPLEX ALGORITHM 141 Let
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6.3. THE SIMPLEX ALGORITHM 143 the
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6.3. THE SIMPLEX ALGORITHM 145 by 2
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6.3. THE SIMPLEX ALGORITHM 147 I ne
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6.3. THE SIMPLEX ALGORITHM 149 Of c
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6.4. FINDING A BASIC FEASIBLE SOLUT
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6.5. DUALITY 153 Now recall that to
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6.5. DUALITY 155 and there are stil
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Spectral Theory Spectral Theory ref
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7.1. EIGENVALUES AND EIGENVECTORS O
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7.2. SOME APPLICATIONS OF EIGENVALU
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7.3. EXERCISES 167 Therefore, the e
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7.3. EXERCISES 169 22. Find the com
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7.3. EXERCISES 171 38. Let M be an
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7.4. SCHUR’S THEOREM 173 7.4 Schu
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7.4. SCHUR’S THEOREM 175 Proof: L
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7.4. SCHUR’S THEOREM 177 and furt
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7.4. SCHUR’S THEOREM 179 Proof: F
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7.6. QUADRATIC FORMS 181 7.6 Quadra
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7.7. SECOND DERIVATIVE TEST 183 Cor
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7.7. SECOND DERIVATIVE TEST 185 The
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7.9. ADVANCED THEOREMS 187 for find
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7.9. ADVANCED THEOREMS 189 Proof: D
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7.10. EXERCISES 191 Show that the a
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7.10. EXERCISES 193 27. Let f (x, y
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7.10. EXERCISES 195 40. In the abov
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7.10. EXERCISES 197 44. Show that i
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Vector Spaces And Fields 8.1 Vector
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8.2. SUBSPACES AND BASES 201 8.2.2
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8.2. SUBSPACES AND BASES 203 Defini
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8.3. LOTS OF FIELDS 205 such that 0
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8.3. LOTS OF FIELDS 207 and this is
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8.3. LOTS OF FIELDS 209 Lemma 8.3.9
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8.3. LOTS OF FIELDS 211 Definition
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8.3. LOTS OF FIELDS 213 Then, since
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8.3. LOTS OF FIELDS 215 Proposition
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8.3. LOTS OF FIELDS 217 where deg r
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8.4. EXERCISES 219 8.3.4 The Lindem
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8.4. EXERCISES 221 18. If you have
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8.4. EXERCISES 223 first is not har
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Linear Transformations 9.1 Matrix M
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9.3. THE MATRIX OF A LINEAR TRANSFO
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9.5. EXERCISES 243 9. ↑In the sit
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Linear Transformations Canonical Fo
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10.1. A THEOREM OF SYLVESTER, DIREC
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10.2. DIRECT SUMS, BLOCK DIAGONAL M
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10.3. CYCLIC SETS 251 while ⎛ ⎞
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10.3. CYCLIC SETS 253 Since β x is
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10.4. NILPOTENT TRANSFORMATIONS 255
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10.5. THE JORDAN CANONICAL FORM 257
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10.6. EXERCISES 263 8. Let ⎛ A =
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10.6. EXERCISES 265 Now use this in
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10.7. THE RATIONAL CANONICAL FORM 2
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10.8. UNIQUENESS 269 Thus the matri
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10.8. UNIQUENESS 271 The characteri
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10.9. EXERCISES 273 Then doing the
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Markov Chains And Migration Process
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11.1. REGULAR MARKOV MATRICES 277 O
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11.2. MIGRATION MATRICES 279 11.2 M
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11.3. MARKOV CHAINS 281 After many
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11.3. MARKOV CHAINS 283 and the fac
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11.4. EXERCISES 285 5. Show that wh
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Inner Product Spaces 12.1 General T
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12.2. THE GRAM SCHMIDT PROCESS 289
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12.2. THE GRAM SCHMIDT PROCESS 291
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12.3. RIESZ REPRESENTATION THEOREM
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12.4. THE TENSOR PRODUCT OF TWO VEC
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12.5. LEAST SQUARES 297 Lemma 12.5.
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12.7. EXERCISES 299 4. Let ||x||
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12.7. EXERCISES 301 (√ √ ) 2 C
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12.8. THE DETERMINANT AND VOLUME 30
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12.8. THE DETERMINANT AND VOLUME 30
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Self Adjoint Operators 13.1 Simulta
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13.1. SIMULTANEOUS DIAGONALIZATION
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13.2. SCHUR’S THEOREM 311 Proof:
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13.3. SPECTRAL THEORY OF SELF ADJOI
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13.3. SPECTRAL THEORY OF SELF ADJOI
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13.4. POSITIVE AND NEGATIVE LINEAR
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13.5. FRACTIONAL POWERS 319 Proof:
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13.5. FRACTIONAL POWERS 321 Proof:
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13.6. POLAR DECOMPOSITIONS 323 Proo
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13.7. AN APPLICATION TO STATISTICS
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13.8. THE SINGULAR VALUE DECOMPOSIT
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13.9. APPROXIMATION IN THE FROBENIU
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13.10. LEAST SQUARES AND SINGULAR V
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13.11. THE MOORE PENROSE INVERSE 33
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13.12. EXERCISES 335 8. Prove the C
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Norms For Finite Dimensional Vector
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339 This proves the first half of t
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341 Next consider the claim that ||
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14.1. THE P NORMS 343 14.1 The p No
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14.2. THE CONDITION NUMBER 345 Thus
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14.2. THE CONDITION NUMBER 347 Sinc
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14.3. THE SPECTRAL RADIUS 349 ⎛ =
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14.4. SERIES AND SEQUENCES OF LINEA
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14.4. SERIES AND SEQUENCES OF LINEA
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14.5. ITERATIVE METHODS FOR LINEAR
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14.6. THEORY OF CONVERGENCE 361 Lem
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14.7. EXERCISES 363 Proof: From Lem
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14.7. EXERCISES 365 11. Suppose ρ
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14.7. EXERCISES 367 Next take e tλ
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14.7. EXERCISES 369 24. Suppose A i
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Numerical Methods For Finding Eigen
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15.1. THE POWER METHOD FOR EIGENVAL
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15.2. THE QR ALGORITHM 387 each a v
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15.2. THE QR ALGORITHM 389 Proof: L
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15.2. THE QR ALGORITHM 391 where
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15.2. THE QR ALGORITHM 393 Corollar
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15.2. THE QR ALGORITHM 395 below th
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15.2. THE QR ALGORITHM 397 where R
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15.2. THE QR ALGORITHM 399 A matrix
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15.3. EXERCISES 401 where A k = Q k
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Positive Matrices Earlier theorems
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405 |μ| ≤λ 0 . But also, the fa
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407 which says x 0 −x 0 v T x 0 =
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409 must have the same argument for
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Functions Of Matrices The existence
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413 There is no convergence problem
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415 This gives us a nice definition
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Applications To Differential Equati
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C.3. LOCAL SOLUTIONS 419 C.3 Local
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C.4. FIRST ORDER LINEAR SYSTEMS 421
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C.5. GEOMETRIC THEORY OF AUTONOMOUS
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C.5. GEOMETRIC THEORY OF AUTONOMOUS
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C.6. GENERAL GEOMETRIC THEORY 433 T
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C.7. THESTABLEMANIFOLD 435 Lemma C.
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C.7. THESTABLEMANIFOLD 437 ≤ e γ
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Compactness And Completeness D.0.1
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441 |a k − a l | < ε 2 .Thenifk>
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The Fundamental Theorem Of Algebra
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Fields And Field Extensions F.1 The
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F.2. THE FUNDAMENTAL THEOREM OF ALG
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F.2. THE FUNDAMENTAL THEOREM OF ALG
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F.3. TRANSCENDENTAL NUMBERS 451 F.3
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F.3. TRANSCENDENTAL NUMBERS 453 It
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F.3. TRANSCENDENTAL NUMBERS 455 Not
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F.3. TRANSCENDENTAL NUMBERS 457 Thi
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F.4. MORE ON ALGEBRAIC FIELD EXTENS
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F.5. THE GALOIS GROUP 465 a rationa
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F.5. THE GALOIS GROUP 467 Now let
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F.6. NORMAL SUBGROUPS 469 If H is a
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F.8. CONDITIONS FOR SEPARABILITY 47
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F.8. CONDITIONS FOR SEPARABILITY 47
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F.9. PERMUTATIONS 475 of f (x) ands
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F.9. PERMUTATIONS 477 smallest k
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F.10. SOLVABLE GROUPS 479 Then i 1
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F.10. SOLVABLE GROUPS 481 Thus ever
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F.11. SOLVABILITY BY RADICALS 483 A
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Bibliography [1] Apostol T., Calcul
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Answers To Selected Exercises G.1 E
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G.7. EXERCISES 489 15 Find the matr
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G.10. EXERCISES 491 11 It follows f
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G.13. EXERCISES 493 19 ⎛ ⎝ 4
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G.19. EXERCISES 495 8 λ 3 − λ 2
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G.23. EXERCISES 497 ⎧⎛ ⎞⎫
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INDEX 499 defective, 162 DeMoivre i
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INDEX 501 rotation, 235 linear tran
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INDEX 503 scalars, 18, 32, 37 Schur
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