Linear Algebra, Theory And Applications, 2012a

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382 NUMERICAL METHODS FOR FINDING EIGENVALUES s 2 = . 66 + . 62i. ⎛ u 2 = ⎝ . 804 878 05 + . 243 902 44i 1.0 . 756 097 56 − . 195 121 95i ⎞ ⎠ = ⎛ ⎝ ⎛ ⎝ ⎛ ⎝ −.02− . 14i 1. 24 + . 68i −. 84 + . 12i −. 14 + .02i .68 − . 24i .12 + . 84i .02+. 14i −. 24 − . 68i .84 + . 88i . 804 878 05 + . 243 902 44i 1.0 . 756 097 56 − . 195 121 95i ⎞ ⎠ . 646 341 46 + . 817 073 17i . 817 073 17 + . 353 658 54i . 548 780 49 − 6. 097 560 9 × 10 −2 i ⎞ ⎠ ⎞ ⎠ · s 3 = . 646 341 46+. 817 073 17i. After more iterations, of this sort, you find s 9 =1. 002 748 5+ 2. 137 621 7 × 10 −4 i and ⎛ u 9 = ⎝ 1.0 . 501 514 17 − . 499 807 33i 1. 562 088 1 × 10 −3 − . 499 778 55i ⎞ ⎠ . Then = ⎛ ⎝ ⎛ ⎝ ⎛ ⎝ −.02− . 14i 1. 24 + . 68i −. 84 + . 12i −. 14 + .02i .68 − . 24i .12 + . 84i .02+. 14i −. 24 − . 68i .84 + . 88i 1.0 . 501 514 17 − . 499 807 33i 1. 562 088 1 × 10 −3 − . 499 778 55i 1. 000 407 8 + 1. 269 979 × 10 −3 i . 501 077 31 − . 498 893 66i 8. 848 928 × 10 −4 − . 499 515 22i ⎞ ⎠ ⎞ ⎠ ⎞ ⎠ · s 10 =1. 000 407 8 + 1. 269 979 × 10 −3 i. ⎛ 1.0 ⎞ u 10 = ⎝ . 500 239 18 − . 499 325 33i 2. 506 749 2 × 10 −4 − . 499 311 92i ⎠ The scaling factors are not changing much at this point. Thus you would solve the following for λ. 1. 000 407 8 + 1. 269 979 × 10 −3 i = 1 λ − i The approximate eigenvalue is then λ = . 999 590 76 + . 998 731 06i. This is pretty close to 1+i. How well does the eigenvector work? = ⎛ ⎝ 5 −8 6 1 0 0 0 1 0 ⎛ ⎝ ⎞ ⎛ ⎠ ⎝ . 999 590 61 + . 998 731 12i 1.0 . 500 239 18 − . 499 325 33i 1.0 . 500 239 18 − . 499 325 33i 2. 506 749 2 × 10 −4 − . 499 311 92i ⎞ ⎠ ⎞ ⎠
15.1. THE POWER METHOD FOR EIGENVALUES 383 = ⎛ (. 999 590 76 + . 998 731 06i) ⎝ ⎛ ⎝ . 999 590 76 + . 998 731 06i . 998 726 18 + 4. 834 203 9 × 10 −4 i . 498 928 9 − . 498 857 22i 1.0 . 500 239 18 − . 499 325 33i 2. 506 749 2 × 10 −4 − . 499 311 92i ⎞ ⎠ ⎞ ⎠ It took more iterations than before because α was not very close to 1 + i. This illustrates an interesting topic which leads to many related topics. If you have a polynomial, x 4 + ax 3 + bx 2 + cx + d, you can consider it as the characteristic polynomial of a certain matrix, called a companion matrix. In this case, ⎛ ⎜ ⎝ −a −b −c −d 1 0 0 0 0 1 0 0 0 0 1 0 ⎞ ⎟ ⎠ . The above example was just a companion matrix for λ 3 − 5λ 2 +8λ − 6. You can see the pattern which will enable you to obtain a companion matrix for any polynomial of the form λ n + a 1 λ n−1 + ···+ a n−1 λ + a n . This illustrates that one way to find the complex zeros of a polynomial is to use the shifted inverse power method on a companion matrix for the polynomial. Doubtless there are better ways but this does illustrate how impressive this procedure is. Do you have a better way? Note that the shifted inverse power method is a way you can begin with something close but not equal to an eigenvalue and end up with something close to an eigenvector. 15.1.4 Rayleigh Quotients <strong>And</strong> Estimates for Eigenvalues There are many specialized results concerning the eigenvalues and eigenvectors for Hermitian matrices. Recall a matrix A is Hermitian if A = A ∗ where A ∗ means to take the transpose of the conjugate of A. In the case of a real matrix, Hermitian reduces to symmetric. Recall also that for x ∈ F n , n∑ |x| 2 = x ∗ x = |x j | 2 . Recall the following corollary found on Page 179 which is stated here for convenience. Corollary 15.1.8 If A is Hermitian, then all the eigenvalues of A are real and there exists an orthonormal basis of eigenvectors. Thus for {x k } n k=1 this orthonormal basis, x ∗ i x j = δ ij ≡ j=1 { 1ifi = j 0ifi ≠ j For x ∈ F n , x ≠ 0, the Rayleigh quotient is defined by x ∗ Ax |x| 2 .
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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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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
Page 363 and 364: 14.7. EXERCISES 363 Proof: From Lem
Page 365 and 366: 14.7. EXERCISES 365 11. Suppose ρ
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 381: 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
Page 429 and 430: C.5. GEOMETRIC THEORY OF AUTONOMOUS
Page 431 and 432: 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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