Linear Algebra, Theory And Applications, 2012a

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384 NUMERICAL METHODS FOR FINDING EIGENVALUES Now let the eigenvalues of A be λ 1 ≤ λ 2 ≤ ··· ≤ λ n and Ax k = λ k x k where {x k } n k=1 is the above orthonormal basis of eigenvectors mentioned in the corollary. Then if x is an arbitrary vector, there exist constants, a i such that n∑ x = a i x i . Also, Therefore, |x| 2 = n∑ a i x ∗ i i=1 = ∑ ij x ∗ Ax |x| 2 = = i=1 n∑ a j x j j=1 a i a j x ∗ i x j = ∑ ij a i a j δ ij = n∑ |a i | 2 . i=1 ( ∑ ( n i=1 a ix ∗ i ) ∑n ) j=1 a jλ j x j ∑ n i=1 |a i| 2 ∑ ij a ia j λ j x ∗ i x ∑ j ij ∑ n i=1 |a i| 2 = a ia j λ j δ ij ∑ n i=1 |a i| 2 = ∑ n i=1 |a i| 2 λ i ∑ n i=1 |a i| 2 ∈ [λ 1 ,λ n ] . In other words, the Rayleigh quotient is always between the largest and the smallest eigenvalues of A. When x = x n , the Rayleigh quotient equals the largest eigenvalue and when x = x 1 the Rayleigh quotient equals the smallest eigenvalue. Suppose you calculate a Rayleigh quotient. How close is it to some eigenvalue? Theorem 15.1.9 Let x ≠ 0 and form the Rayleigh quotient, x ∗ Ax |x| 2 ≡ q. Then there exists an eigenvalue of A, denoted here by λ q such that |Ax − qx| |λ q − q| ≤ . (15.6) |x| Proof: Let x = ∑ n k=1 a kx k where {x k } n k=1 is the orthonormal basis of eigenvectors. |Ax − qx| 2 = (Ax − qx) ∗ (Ax − qx) ( n ) ∗ ( ∑ n ) ∑ = a k λ k x k − qa k x k a k λ k x k − qa k x k k=1 k=1 ⎛ ⎞ ( n∑ n ) ∑ = ⎝ (λ j − q) a j x ∗ ⎠ j (λ k − q) a k x k j=1 k=1 = ∑ j,k (λ j − q) a j (λ k − q) a k x ∗ j x k = n∑ |a k | 2 (λ k − q) 2 k=1
15.1. THE POWER METHOD FOR EIGENVALUES 385 Now pick the eigenvalue λ q which is closest to q. Then |Ax − qx| 2 = n∑ ∑ n |a k | 2 (λ k − q) 2 ≥ (λ q − q) 2 |a k | 2 =(λ q − q) 2 |x| 2 k=1 which implies (15.6). Example 15.1.10 Consider the symmetric matrix A = ⎛ ⎝ 1 2 2 2 3 1 ⎞ ⎠ . Let x =(1, 1, 1) T . 3 1 4 How close is the Rayleigh quotient to some eigenvalue of A? Find the eigenvector and eigenvalue to several decimal places. Everything is real and so there is no need to worry about taking conjugates. Therefore, the Rayleigh quotient is ⎛ ( ) 1 1 1 ⎝ 1 2 3 ⎞ ⎛ 2 2 1 ⎠ ⎝ 1 ⎞ 1 ⎠ 3 1 4 1 = 19 3 3 According to the above theorem, there is some eigenvalue of this matrix λ q such that ∣⎛ ⎞ ⎛ ⎞ ⎛ ⎞∣ ∣ λ q − 19 3 ∣ ⎝ 1 2 3 2 2 1 ⎠ ⎝ 1 1 ∣ 3 1 4 1 ≤ √ 3 ⎛ ⎞ = √ 1 − 1 3 ⎝ − 4 ⎠ 3 3 5 3 = √ 1 ) 2 k=1 ⎠ − 19 3 9 + ( 4 2 ( 3) + 5 3 √ =1. 247 2 3 ⎝ 1 1 1 Could you find this eigenvalue and associated eigenvector? Of course you could. This is what the shifted inverse power method is all about. Solve ⎛⎛ ⎝⎝ 1 2 3 ⎞ ⎛ 2 2 1 ⎠ − 19 ⎝ 1 0 0 ⎞⎞ ⎛ 0 1 0 ⎠⎠ ⎝ x ⎞ ⎛ y ⎠ = ⎝ 1 ⎞ 1 ⎠ 3 3 1 4 0 0 1 z 1 In other words solve ⎛ ⎞ ⎛ − 16 3 2 3 ⎝ 2 − 13 3 1 ⎠ ⎝ x ⎞ y ⎠ = 3 1 − 7 z 3 and divide by the entry which is largest, 3. 870 7, to get ⎛ u 2 = ⎝ . 699 25 . 493 89 1.0 ⎞ ⎠ ⎛ ⎝ 1 1 1 ⎞ ⎠ ⎠ ∣ Now solve ⎛ ⎝ − 16 3 2 3 2 − 13 3 1 3 1 − 7 3 ⎞ ⎛ ⎠ ⎝ x y z ⎞ ⎛ ⎠ = ⎝ . 699 25 . 493 89 1.0 ⎞ ⎠
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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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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 383: 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
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Page 395 and 396: 15.2. THE QR ALGORITHM 395 below th
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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
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
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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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