Linear Algebra, Theory And Applications, 2012a

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378 NUMERICAL METHODS FOR FINDING EIGENVALUES because all these are only approximations to the eigenvalue and so the matrix in the above is nonsingular for all of these. Therefore, you will only get the zero solution and Eigenvectors are never equal to zero! However, there exists such an eigenvector and you can find it using the shifted inverse power method. Pick α = −.855. Then you solve ⎛⎛ ⎝⎝ 1 2 3 ⎞ ⎛ 2 1 4 ⎠ + .855 ⎝ 1 0 0 ⎞⎞ ⎛ 0 1 0 ⎠⎠ ⎝ x ⎞ ⎛ y ⎠ = ⎝ 1 ⎞ 1 ⎠ 3 4 2 0 0 1 z 1 or in other words, ⎛ ⎝ 1. 855 2.0 3.0 2.0 1. 855 4.0 3.0 4.0 2. 855 ⎞ ⎛ ⎞ ⎛ ⎞ ⎠ ⎝ x y z ⎠ = ⎝ 1 1 ⎠ 1 and after finding the solution, divide by the largest entry −67. 944, to obtain ⎛ u 2 = ⎝ 1. 0 ⎞ −. 589 21 ⎠ −. 230 44 After a couple more iterations, you obtain ⎛ u 3 = ⎝ 1. 0 ⎞ −. 587 77 ⎠ (15.5) −. 227 14 Then doing it again, the scaling factor is −513. 42 and the next iterate is ⎛ u 4 = ⎝ 1. 0 ⎞ −. 587 78 ⎠ −. 227 14 Clearly the u k are not changing much. This suggests an approximate eigenvector for this eigenvalue which is close to −.855 is the above u 3 and an eigenvalue is obtained by solving 1 = −514. 01, λ + .855 which yields λ = −. 856 9 Lets check this. ⎛ ⎝ 1 2 3 ⎞ ⎛ 2 1 4 ⎠ ⎝ 1. 0 ⎞ ⎛ −. 587 77 ⎠ = ⎝ 3 4 2 −. 227 14 ⎛ −. 856 9 ⎝ 1. 0 −. 587 77 −. 227 14 ⎞ ⎛ ⎠ = ⎝ −. 856 96 . 503 67 . 194 64 −. 856 9 . 503 7 . 194 6 Thus the vector of (15.5) is very close to the desired eigenvector, just as −. 856 9 is very close to the desired eigenvalue. For practical purposes, I have found both the eigenvector and the eigenvalue. ⎞ ⎠ ⎞ ⎠ .
15.1. THE POWER METHOD FOR EIGENVALUES 379 Example 15.1.6 Find the eigenvalues and eigenvectors of the matrix A = ⎝ ⎛ 2 1 3 2 1 1 3 2 1 This is only a 3×3 matrix and so it is not hard to estimate the eigenvalues. Just get the characteristic equation, graph it using a calculator and zoom in to find the eigenvalues. If you do this, you find there is an eigenvalue near −1.2, one near −.4, and one near 5.5. (The characteristic equation is 2 + 8λ +4λ 2 − λ 3 =0.) Of course I have no idea what the eigenvectors are. Lets first try to find the eigenvector and a better approximation for the eigenvalue near −1.2. In this case, let α = −1.2. Then ⎛ (A − αI) −1 = ⎝ −25. 357 143 −33. 928 571 50.0 12. 5 17. 5 −25.0 23. 214 286 30. 357 143 −45.0 As before, it helps to get things started if you raise to a power and then go from the approximate eigenvector obtained. ⎛ ⎞7 ⎛ −25. 357 143 −33. 928 571 50.0 ⎝ 12. 5 17. 5 −25.0 ⎠ ⎝ 1 ⎞ ⎛ ⎞ −2. 295 6 × 1011 1 ⎠ = ⎝ 1. 129 1 × 10 11 ⎠ 23. 214 286 30. 357 143 −45.0 1 2. 086 5 × 10 11 Then the next iterate will be ⎛ ⎝ ⎞ −2. 295 6 × 1011 1. 129 1 × 10 11 ⎠ ⎛ 1 −2. 295 6 × 10 11 = ⎝ Next iterate: ⎛ ⎞ ⎛ −25. 357 143 −33. 928 571 50.0 ⎝ 12. 5 17. 5 −25.0 ⎠ ⎝ 23. 214 286 30. 357 143 −45.0 Divide by largest entry ⎛ ⎝ −54. 115 26. 615 49. 184 ⎞ ⎠ ⎛ 1 −54. 115 = ⎝ 1.0 −0.491 85 −0.908 91 1.0 −0.491 82 −0.908 88 1.0 −0.491 85 −0.908 91 ⎞ ⎛ ⎠ = ⎝ ⎞ ⎠ ⎞ ⎠ . ⎞ ⎠ −54. 115 26. 615 49. 184 You can see the vector didn’t change much and so the next scaling factor will not be much different than this one. Hence you need to solve for λ 1 = −54. 115 λ +1.2 Then λ = −1. 218 5 is an approximate eigenvalue and ⎛ ⎝ 1.0 −0.491 82 −0.908 88 ⎞ ⎠ is an approximate eigenvector. How well does it work? ⎛ ⎝ 2 1 3 ⎞ ⎛ ⎞ 1.0 2 1 1 ⎠ ⎝ −0.491 82 ⎠ = 3 2 1 −0.908 88 ⎛ (−1. 218 5) ⎝ 1.0 −0.491 82 −0.908 88 ⎞ ⎠ = ⎛ ⎝ ⎛ ⎝ −1. 218 5 0.599 3 1. 107 5 −1. 218 5 0.599 28 1. 107 5 ⎞ ⎠ ⎞ ⎠ ⎞ ⎠ ⎞ ⎠ .
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Linear Algebra, Theory And Applicat
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Contents 1 Preliminaries 11 1.1 Set
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CONTENTS 5 8 Vector Spaces And Fiel
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CONTENTS 7 E The Fundamental Theore
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Preface This is a book on linear al
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Preliminaries 1.1 Sets And Set Nota
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1.3. THE NUMBER LINE AND ALGEBRA OF
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1.5. THE COMPLEX NUMBERS 15 Also fr
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1.5. THE COMPLEX NUMBERS 17 and so
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1.6. EXERCISES 19 Eventually, for r
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1.8. WELL ORDERING AND ARCHIMEDEAN
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1.9. DIVISION AND NUMBERS 23 Theore
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1.9. DIVISION AND NUMBERS 25 Exampl
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1.10. SYSTEMS OF EQUATIONS 27 Why i
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1.10. SYSTEMS OF EQUATIONS 29 The a
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1.11. EXERCISES 31 It follows x =
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1.14. EXERCISES 33 the existence of
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1.15. THE INNER PRODUCT IN F N 35 s
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Matrices And Linear Transformations
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2.1. MATRICES 39 Definition 2.1.2 M
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2.1. MATRICES 41 is it possible to
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2.1. MATRICES 43 a3× 3 matrix as d
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2.1. MATRICES 45 1 2 3 4 Write the
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= ∑ k ( B T ) ik ( A T ) kj 2.1.
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2.1. MATRICES 49 As described above
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2.2. EXERCISES 51 Write the augment
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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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Page 335 and 336: 13.12. EXERCISES 335 8. Prove the C
Page 337 and 338: Norms For Finite Dimensional Vector
Page 339 and 340: 339 This proves the first half of t
Page 341 and 342: 341 Next consider the claim that ||
Page 343 and 344: 14.1. THE P NORMS 343 14.1 The p No
Page 345 and 346: 14.2. THE CONDITION NUMBER 345 Thus
Page 347 and 348: 14.2. THE CONDITION NUMBER 347 Sinc
Page 349 and 350: 14.3. THE SPECTRAL RADIUS 349 ⎛ =
Page 351 and 352: 14.4. SERIES AND SEQUENCES OF LINEA
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Page 355 and 356: 14.5. ITERATIVE METHODS FOR LINEAR
Page 361 and 362: 14.6. THEORY OF CONVERGENCE 361 Lem
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Page 367 and 368: 14.7. EXERCISES 367 Next take e tλ
Page 369 and 370: 14.7. EXERCISES 369 24. Suppose A i
Page 371 and 372: Numerical Methods For Finding Eigen
Page 373 and 374: 15.1. THE POWER METHOD FOR EIGENVAL
Page 377: 15.1. THE POWER METHOD FOR EIGENVAL
Page 387 and 388: 15.2. THE QR ALGORITHM 387 each a v
Page 389 and 390: 15.2. THE QR ALGORITHM 389 Proof: L
Page 391 and 392: 15.2. THE QR ALGORITHM 391 where
Page 393 and 394: 15.2. THE QR ALGORITHM 393 Corollar
Page 395 and 396: 15.2. THE QR ALGORITHM 395 below th
Page 397 and 398: 15.2. THE QR ALGORITHM 397 where R
Page 399 and 400: 15.2. THE QR ALGORITHM 399 A matrix
Page 401 and 402: 15.3. EXERCISES 401 where A k = Q k
Page 403 and 404: Positive Matrices Earlier theorems
Page 405 and 406: 405 |μ| ≤λ 0 . But also, the fa
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Page 411 and 412: Functions Of Matrices The existence
Page 413 and 414: 413 There is no convergence problem
Page 415 and 416: 415 This gives us a nice definition
Page 417 and 418: Applications To Differential Equati
Page 419 and 420: C.3. LOCAL SOLUTIONS 419 C.3 Local
Page 421 and 422: 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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