Linear Algebra, Theory And Applications, 2012a

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400 NUMERICAL METHODS FOR FINDING EIGENVALUES Let A be an invertible n × n matrix. Let Q ′ 1 be a unitary matrix ⎛ √ ⎛ ⎞ ∑n ⎞ a 21 j=2 |a j1| 2 ⎛ ⎞ a Q ′ ⎜ . ⎟ 1 ⎝ ⎠ = 0 ⎜ ⎝ ⎟ a n1 . ⎠ ≡ 0 ⎜ ⎟ ⎝ . ⎠ 0 0 The vector Q ′ 1 is multiplying is just the bottom n − 1 entries of the first column of A. Then let Q 1 be ( 1 ) 0 0 Q ′ 1 It follows ( Q 1 AQ ∗ 1 0 1 = 0 Q ′ 1 ) AQ ∗ 1 = ⎛ = ⎜ ⎝ ⎛ ⎜ ⎝ a 11 a 12 ··· a 1n a . A ′ 1 0 ∗ ∗ ··· ∗ a . A 1 0 ⎞ ⎟ ⎠ ⎞ ( 1 0 ⎟ ⎠ 0 Q ′∗ 1 Now let Q ′ 2 be the n − 2 × n − 2 matrix which does to the first column of A 1 the same sort of thing that the n − 1 × n − 1 matrix Q ′ 1 did to the first column of A. Let ( ) I 0 Q 2 ≡ 0 Q ′ 2 where I is the 2 × 2 identity. Then applying block multiplication, ⎛ ⎞ ∗ ∗ ··· ∗ ∗ ∗ ∗ ··· ∗ ∗ Q 2 Q 1 AQ ∗ 1Q ∗ 2 = 0 ∗ ⎜ ⎝ . ⎟ . A 2 ⎠ 0 0 where A 2 is now an n − 2 × n − 2 matrix. Continuing this way you eventually get a unitary matrix Q which is a product of those discussed above such that ⎛ ⎞ ∗ ∗ ··· ∗ ∗ ∗ ∗ ··· ∗ ∗ QAQ T . = 0 ∗ ∗ . ⎜ . . ⎝ . . . .. . ⎟ .. ∗ ⎠ 0 0 ∗ ∗ This matrix equals zero below the subdiagonal. It is called an upper Hessenberg matrix. It happens that in the QR algorithm, if A k is upper Hessenberg, so is A k+1 . To see this, note that the matrix is upper Hessenberg means that A ij = 0 whenever i − j ≥ 2. A k+1 = R k Q k )
15.3. EXERCISES 401 where A k = Q k R k . Therefore as shown before, A k+1 = R k A k R −1 k Let the ij th entry of A k be a k ij .Thenifi − j ≥ 2 a k+1 ij = n∑ j∑ p=i q=1 r ip a k pqr −1 qj It is given that a k pq = 0 whenever p − q ≥ 2. However, from the above sum, p − q ≥ i − j ≥ 2 and so the sum equals 0. Since upper Hessenberg matrices stay that way in the algorithm and it is closer to being upper triangular, it is reasonable to suppose the QR algorithm will yield good results more quickly for this upper Hessenberg matrix than for the original matrix. This would be especially true if the matrix is good sized. The other important thing to observe is that, starting with an upper Hessenberg matrix, the algorithm will restrict the size of the blocks which occur to being 2 × 2 blocks which are easy to deal with. These blocks allow you to identify the complex roots. 15.3 Exercises In these exercises which call for a computation, don’t waste time on them unless you use a computer or calculator which can raise matrices to powers and take QR factorizations. 1. In Example 15.1.10 an eigenvalue was found correct to several decimal places along with an eigenvector. Find the other eigenvalues along with their eigenvectors. 2. Find the eigenvalues and eigenvectors of the matrix A = ⎛ ⎝ 3 2 2 1 1 3 ⎞ ⎠ numerically. 1 3 2 In this case the exact eigenvalues are ± √ 3, 6. Compare with the exact answers. ⎛ 3 2 1 ⎞ 3. Find the eigenvalues and eigenvectors of the matrix A = ⎝ 2 5 3 ⎠ numerically. 1 3 2 The exact eigenvalues are 2, 4+ √ 15, 4 − √ 15. Compare your numerical results with the exact values. Is it much fun to compute the exact eigenvectors? ⎛ 0 2 1 ⎞ 4. Find the eigenvalues and eigenvectors of the matrix A = ⎝ 2 5 3 ⎠ numerically. 1 3 2 I don’t know the exact eigenvalues in this case. Check your answers by multiplying your numerically computed eigenvectors by the matrix. 5. Find the eigenvalues and eigenvectors of the matrix A = ⎛ ⎝ 0 2 2 0 1 3 ⎞ ⎠ numerically. 1 3 2 I don’t know the exact eigenvalues in this case. Check your answers by multiplying your numerically computed eigenvectors by the matrix.
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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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14.2. THE CONDITION NUMBER 347 Sinc
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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
Page 397 and 398: 15.2. THE QR ALGORITHM 397 where R
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Page 413 and 414: 413 There is no convergence problem
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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
Page 449 and 450: 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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