Linear Algebra, Theory And Applications, 2012a

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392 NUMERICAL METHODS FOR FINDING EIGENVALUES that ΛU is an upper triangular matrix having positive diagonal entries. Note Λ 2 = I and also Λ commutes with a diagonal matrix. Thus Q (k) R (k) = QQ k R k RD k Λ 2 U = QQ k R k RΛD k (ΛU) At this point, one does some inspired massaging to write the above in the form QQ k ( ΛD k ) [ ( ΛD k ) −1 Rk RΛD k] (ΛU) = Q (Q k Λ) D k [ ( ΛD k ) −1 Rk RΛD k] (ΛU) ≡G k { [ }} { = Q (Q k Λ) D k ( ΛD k ) −1 Rk RΛD k] (ΛU) Now I claim the middle matrix in [·] is upper triangular and has all positive entries on the diagonal. This is because it is an upper triangular matrix which is similar to the upper triangular matrix R k R[ and so it has the same eigenvalues (diagonal entries) as R k R.Thus (ΛD the matrix G k ≡ D ) ] k k −1 Rk RΛD k (ΛU) is upper triangular and has all positive entries on the diagonal. Multiply on the right by G −1 k to get Q (k) R (k) G −1 k = QQ k Λ → Q ′ where Q ′ is essentially equal to Q but might have some of the columns multiplied by −1. This is because Q k → I and so Q k Λ → Λ. Now by Lemma 15.2.3, it follows Q (k) → Q ′ ,R (k) G −1 k → I. It remains to verify A k converges to an upper triangular matrix. Recall that from (15.12) and the definition below this (S = QR) A = SDS −1 =(QR) D (QR) −1 = QRDR −1 Q T = QT Q T Where T is an upper triangular matrix. This is because it is the product of upper triangular matrices R, D, R −1 .Thus Q T AQ = T. If you replace Q with Q ′ in the above, it still results in an upper triangular matrix T ′ having the same diagonal entries as T. This is because T = Q T AQ =(Q ′ Λ) T A (Q ′ Λ) = ΛQ ′T AQ ′ Λ and considering the ii th entry yields ( Q T AQ ) ii ≡ ∑ ( Λ ij Q ′T AQ ′) jk Λ ( ki =Λ ii Λ ii Q ′T AQ ′) ii = ( Q ′T AQ ′) ii j,k Recall from Lemma 15.2.2, Thus taking a limit and using the first part, A k = Q (k)T AQ (k) A k = Q (k)T AQ (k) → Q ′T AQ ′ = T ′ . An easy case is for A symmetric. Recall Corollary 7.4.13. By this corollary, there exists an orthogonal (real unitary) matrix Q such that Q T AQ = D where D is diagonal having the eigenvalues on the main diagonal decreasing in size from the upper left corner to the lower right.
15.2. THE QR ALGORITHM 393 Corollary 15.2.5 Let A be a real symmetric n × n matrix having eigenvalues λ 1 >λ 2 > ···>λ n > 0 and let Q be defined by QDQ T = A, D = Q T AQ, (15.14) where Q is orthogonal and D is a diagonal matrix having the eigenvalues on the main diagonal decreasing in size from the upper left corner to the lower right. Let Q T have an LU factorization. Then in the QR algorithm, the matrices Q (k) converge to Q ′ where Q ′ is thesameasQ except having some columns multiplied by (−1) . Thus the columns of Q ′ are eigenvectors of A. The matrices A k converge to D. Proof: This follows from Theorem 15.2.4. Here S = Q, S −1 = Q T . Thus Q = S = QR and R = I. By Theorem 15.2.4 and Lemma 15.2.2, A k = Q (k)T AQ (k) → Q ′T AQ ′ = Q T AQ = D. because formula (15.14) is unaffected by replacing Q with Q ′ . When using the QR algorithm, it is not necessary to check technical condition about S −1 having an LU factorization. The algorithm delivers a sequence of matrices which are similar to the original one. If that sequence converges to an upper triangular matrix, then the algorithm worked. Furthermore, the technical condition is sufficient but not necessary. The algorithm will work even without the technical condition. Example 15.2.6 Find the eigenvalues and eigenvectors of the matrix ⎛ A = ⎝ 5 1 1 ⎞ 1 3 2 ⎠ 1 2 1 It is a symmetric matrix but other than that, I just pulled it out of the air. By Lemma 15.2.2 it follows A k = Q (k)T AQ (k) . <strong>And</strong> so to get to the answer quickly I could have the computer raise A toapowerandthentaketheQR factorization of what results to get the k th iteration using the above formula. Lets pick k =10. ⎛ ⎝ 5 1 1 3 1 2 ⎞ ⎠ 1 2 1 10 ⎛ = ⎝ 4. 227 3 × ⎞ 107 2. 595 9 × 10 7 1. 861 1 × 10 7 2. 595 9 × 10 7 1. 607 2 × 10 7 1. 150 6 × 10 7 ⎠ 1. 861 1 × 10 7 1. 150 6 × 10 7 8. 239 6 × 10 6 Now take QR factorization of this. The computer will do that also. This yields ⎛ ⎞ . 797 85 −. 599 12 −6. 694 3 × 10−2 ⎝ . 489 95 . 709 12 −. 507 06 ⎠ · . 351 26 . 371 76 . 859 31 ⎛ ⎝ 5. 298 3 × ⎞ 107 3. 262 7 × 10 7 2. 338 × 10 7 0 1. 217 2 × 10 5 71946. ⎠ 0 0 277. 03
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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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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: 15.2. THE QR ALGORITHM 391 where
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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
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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
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>
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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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