Linear Algebra, Theory And Applications, 2012a

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398 NUMERICAL METHODS FOR FINDING EIGENVALUES How do you identify approximate eigenvectors from this? First try to find the approximate eigenvectors for A k . Pick an approximate eigenvalue λ, an exact eigenvalue for T ′ B . Then find v solving T ′ B v = λv. It follows since T ′ B is close to A k that and so A k v ≈ λv Q (k) AQ (k)T v = A k v ≈ λv Hence AQ (k)T v ≈ λQ (k)T v and so Q (k)T v is an approximation to the eigenvector which goes with the eigenvalue of A which is close to λ. Example 15.2.7 Here is a matrix. ⎛ ⎝ −2 0 −1 3 2 1 −2 −2 0 It happens that the eigenvalues of this matrix are 1, 1+i, 1 − i. Lets apply the QR algorithm as if the eigenvalues were not known. ⎞ ⎠ Applying the QR algorithm to this matrix yields the following sequence of matrices. ⎛ ⎞ 1. 235 3 1. 941 2 4. 365 7 A 1 = ⎝ −. 392 15 1. 542 5 5. 388 6 × 10 −2 ⎠ −. 161 69 −. 188 64 . 222 22 ⎛ A 12 = ⎝ . 9. 177 2 × 10 −2 . 630 89 −2. 039 8 −2. 855 6 1. 908 2 −3. 104 3 1. 078 6 × 10 −2 3. 461 4 × 10 −4 1.0 At this point the bottom two terms on the left part of the bottom row are both very small so it appears the real eigenvalue is near 1.0. The complex eigenvalues are obtained from solving This yields ( ( 1 0 det λ 0 1 ) ( 9. 177 2 × 10 −2 . 630 89 − −2. 855 6 1. 908 2 λ =1.0 − . 988 28i, 1.0+. 988 28i ⎞ ⎠ )) =0 Example 15.2.8 The equation x 4 +x 3 +4x 2 +x−2 =0has exactly two real solutions. You can see this by graphing it. However, the rational root theorem from algebra shows neither of these solutions are rational. Also, graphing it does not yield any information about the complex solutions. Lets use the QR algorithm to approximate all the solutions, real and complex.
15.2. THE QR ALGORITHM 399 A matrix whose characteristic polynomial is the given polynomial is ⎛ ⎞ −1 −4 −1 2 ⎜ 1 0 0 0 ⎟ ⎝ 0 1 0 0 ⎠ 0 0 1 0 Using the QR algorithm yields the following sequence of iterates for A k ⎛ ⎞ . 999 99 −2. 592 7 −1. 758 8 −1. 297 8 A 1 = ⎜ 2. 121 3 −1. 777 8 −1. 604 2 −. 994 15 ⎟ ⎝ 0 . 342 46 −. 327 49 −. 917 99 ⎠ 0 0 −. 446 59 . 105 26 ⎛ A 9 = ⎜ ⎝ . −. 834 12 −4. 168 2 −1. 939 −. 778 3 1. 05 . 145 14 . 217 1 2. 547 4 × 10 −2 0 4. 026 4 × 10 −4 −. 850 29 −. 616 08 0 0 −1. 826 3 × 10 −2 . 539 39 Now this is similar to A and the eigenvalues are close to the eigenvalues obtained from the two blocks on the diagonal. Of course the lower left corner of the bottom block is vanishing but it is still fairly large so the eigenvalues are approximated by the solution to ( ( ) ( )) 1 0 −. 850 29 −. 616 08 det λ − 0 1 −1. 826 3 × 10 −2 =0 . 539 39 The solution to this is λ = −. 858 34, .547 44 and for the complex eigenvalues, The solution is det ( ( 1 0 λ 0 1 ) ( −. 834 12 −4. 168 2 − 1. 05 . 145 14 )) =0 λ = −. 344 49 − 2. 033 9i, −. 344 49 + 2. 033 9i ⎞ ⎟ ⎠ How close are the complex eigenvalues just obtained to giving a solution to the original equation? Try −. 344 49 + 2. 033 9i .When this is plugged in it yields −.00 12 + 2. 006 8 × 10 −4 i which is pretty close to 0. The real eigenvalues are also very close to the corresponding real solutions to the original equation. It seems like most of the attention to the QR algorithm has to do with finding ways to get it to “converge” faster. Great and marvelous are the clever tricks which have been proposed to do this but my intent is to present the basic ideas, not to go in to the numerous refinements of this algorithm. However, there is one thing which is usually done. It involves reducing to the case of an upper Hessenberg matrix which is one which is zero below the main sub diagonal. To see that every matrix is unitarily similar to an upper Hessenberg matrix , see Problem 1 on Page 273. What follows is a construction which also proves this.
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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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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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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 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 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
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Page 395 and 396: 15.2. THE QR ALGORITHM 395 below th
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Page 401 and 402: 15.3. EXERCISES 401 where A k = Q k
Page 403 and 404: Positive Matrices Earlier theorems
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Page 411 and 412: Functions Of Matrices The existence
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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
Page 429 and 430: C.5. GEOMETRIC THEORY OF AUTONOMOUS
Page 431 and 432: C.5. GEOMETRIC THEORY OF AUTONOMOUS
Page 433 and 434: C.6. GENERAL GEOMETRIC THEORY 433 T
Page 435 and 436: C.7. THESTABLEMANIFOLD 435 Lemma C.
Page 437 and 438: C.7. THESTABLEMANIFOLD 437 ≤ e γ
Page 439 and 440: Compactness And Completeness D.0.1
Page 441 and 442: 441 |a k − a l | < ε 2 .Thenifk>
Page 443 and 444: The Fundamental Theorem Of Algebra
Page 445 and 446: Fields And Field Extensions F.1 The
Page 447 and 448: 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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