Linear Algebra, Theory And Applications, 2012a

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4 CONTENTS 3.3.2 The Definition Of The Determinant . . . . . . . . . . . . . . . . . . . 86 3.3.3 A Symmetric Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.3.4 Basic Properties Of The Determinant . . . . . . . . . . . . . . . . . . 88 3.3.5 Expansion Using Cofactors . . . . . . . . . . . . . . . . . . . . . . . . 90 3.3.6 A Formula For The Inverse . . . . . . . . . . . . . . . . . . . . . . . . 92 3.3.7 Rank Of A Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.3.8 Summary Of Determinants . . . . . . . . . . . . . . . . . . . . . . . . 96 3.4 The Cayley Hamilton Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.5 Block Multiplication Of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4 Row Operations 105 4.1 Elementary Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.2 The Rank Of A Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.3 The Row Reduced Echelon Form . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.4 Rank And Existence Of Solutions To Linear Systems . . . . . . . . . . . . . . 116 4.5 Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5 Some Factorizations 123 5.1 LU Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.2 Finding An LU Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.3 Solving Linear Systems Using An LU Factorization . . . . . . . . . . . . . . . 125 5.4 The PLU Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.5 Justification For The Multiplier Method . . . . . . . . . . . . . . . . . . . . . 127 5.6 Existence For The PLU Factorization . . . . . . . . . . . . . . . . . . . . . . 128 5.7 The QR Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6 Linear Programming 135 6.1 Simple Geometric Considerations . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.2 The Simplex Tableau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.3 The Simplex Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.3.1 Maximums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.3.2 Minimums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.4 Finding A Basic Feasible Solution . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.5 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7 Spectral Theory 157 7.1 Eigenvalues And Eigenvectors Of A Matrix . . . . . . . . . . . . . . . . . . . 157 7.2 Some Applications Of Eigenvalues And Eigenvectors . . . . . . . . . . . . . . 164 7.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 7.4 Schur’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 7.5 Trace And Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.6 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.7 Second Derivative Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.8 The Estimation Of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . 186 7.9 Advanced Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
CONTENTS 5 8 Vector Spaces And Fields 199 8.1 Vector Space Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 8.2 Subspaces And Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 8.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 8.2.2 A Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 201 8.2.3 The Basis Of A Subspace . . . . . . . . . . . . . . . . . . . . . . . . . 205 8.3 Lots Of Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 8.3.1 Irreducible Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 205 8.3.2 Polynomials And Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 210 8.3.3 The Algebraic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 215 8.3.4 The Lindemannn Weierstrass Theorem And Vector Spaces . . . . . . . 219 8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 9 Linear Transformations 225 9.1 Matrix Multiplication As A Linear Transformation . . . . . . . . . . . . . . . 225 9.2 L (V,W) As A Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 9.3 The Matrix Of A Linear Transformation . . . . . . . . . . . . . . . . . . . . . 227 9.3.1 Some Geometrically Defined Linear Transformations . . . . . . . . . . 234 9.3.2 Rotations About A Given Vector . . . . . . . . . . . . . . . . . . . . . 237 9.3.3 The Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 9.4 Eigenvalues And Eigenvectors Of Linear Transformations . . . . . . . . . . . 240 9.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 10 Linear Transformations Canonical Forms 245 10.1 A Theorem Of Sylvester, Direct Sums . . . . . . . . . . . . . . . . . . . . . . 245 10.2 Direct Sums, Block Diagonal Matrices . . . . . . . . . . . . . . . . . . . . . . 248 10.3 Cyclic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 10.4 Nilpotent Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 10.5 The Jordan Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 10.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 10.7 The Rational Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 10.8 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 10.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 11 Markov Chains And Migration Processes 275 11.1 Regular Markov Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 11.2 Migration Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 11.3 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 11.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 12 Inner Product Spaces 287 12.1 General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 12.2 The Gram Schmidt Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 12.3 Riesz Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 292 12.4 The Tensor Product Of Two Vectors . . . . . . . . . . . . . . . . . . . . . . . 295 12.5 Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 12.6 Fredholm Alternative Again . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 12.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 12.8 The Determinant And Volume . . . . . . . . . . . . . . . . . . . . . . . . . . 303 12.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
Page 1 and 2: Linear Algebra, Theory And Applicat
Page 3: Contents 1 Preliminaries 11 1.1 Set
Page 7 and 8: CONTENTS 7 E The Fundamental Theore
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)
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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
Page 495 and 496:
G.19. EXERCISES 495 8 λ 3 − λ 2
Page 497 and 498:
G.23. EXERCISES 497 ⎧⎛ ⎞⎫
Page 499 and 500:
INDEX 499 defective, 162 DeMoivre i
Page 501 and 502:
INDEX 501 rotation, 235 linear tran
Page 503:
INDEX 503 scalars, 18, 32, 37 Schur
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Linear Algebra, Theory And Applications, 2012a

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