Linear Algebra, Theory And Applications, 2012a

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108 ROW OPERATIONS Denote by E (c, i) this elementary matrix which multiplies the i th row of the identity by the nonzero constant, c. Then from what was just discussed and the way matrices are multiplied, ⎛ ⎞ a 11 a 12 ··· ··· a 1p . . . E (c, i) a i1 a i2 ··· ··· a ip ⎜ ⎟ ⎝ . . . ⎠ a n1 a n2 ··· ··· a np equals a matrix having the columns indicated below. ⎛ ⎛ ⎞ ⎛ ⎞ ⎛ a 11 a 12 . . = E (c, i) a i1 ,E(c, i) a i2 , ··· ,E(c, i) ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎝ ⎝ . ⎠ ⎝ . ⎠ ⎝ a n1 a n2 ⎛ ⎞ a 11 a 12 ··· ··· a 1p . . . = ca i1 ca i2 ··· ··· ca ip ⎜ ⎟ ⎝ . . . ⎠ a n1 a n2 ··· ··· a np This proves the following lemma. a 1p . . a ip . a np ⎞⎞ ⎟⎟ ⎠⎠ Lemma 4.1.4 Let E (c, i) denote the elementary matrix corresponding to the row operation in which the i th row is multiplied by the nonzero constant, c. Thus E (c, i) involves multiplying the i th row of the identity matrix by c. Then E (c, i) A = B where B is obtained from A by multiplying the i th row of A by c. Finally consider the third of these row operations. Denote by E (c × i + j) the elementary matrix which replaces the j th rowwithitselfaddedtoc times the i th row added to it. In case i
4.1. ELEMENTARY MATRICES 109 Now consider what this does to a column vector. ⎛ ⎞ ⎛ 1 0 . .. 1 . . .. c ··· 1 ⎜ ⎝ . ⎟ ⎜ .. ⎠ ⎝ 0 1 v 1 . . v i . . v j . . v n ⎞ ⎛ = ⎟ ⎜ ⎠ ⎝ v 1 . . v i . . cv i + v j . v n ⎞ ⎟ ⎠ Now from this and the way matrices are multiplied, ⎛ ⎞ a 11 a 12 ··· ··· ··· ··· a 1p . . . a i1 a i2 ··· ··· ··· ··· a ip E (c × i + j) . . . a j2 a j2 ··· ··· ··· ··· a jp ⎜ ⎝ . ⎟ . . ⎠ a n1 a n2 ··· ··· ··· ··· a np equals a matrix of the following form having the indicated columns. ⎛ ⎛ ⎞ ⎛ ⎞ ⎛ a 11 a 12 . . a i1 a i2 E (c × i + j) . ,E(c × i + j) . , ···E (c × i + j) a j2 a j2 ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎝ ⎝ . ⎠ ⎝ . ⎠ ⎝ a n1 a n2 ⎛ = ⎜ ⎝ ⎞ a 11 a 12 ··· a 1p . . . a i1 a i2 ··· a ip . . . . . . a j2 + ca i1 a j2 + ca i2 ··· a jp + ca ip . . . ⎟ . . . ⎠ a n1 a n2 ··· a np Thecasewherei>jis handled similarly. This proves the following lemma. a 1p . . a ip . a jp . a np ⎞⎞ ⎟⎟ ⎠⎠ Lemma 4.1.5 Let E (c × i + j) denote the elementary matrix obtained from I by replacing the j th row with c times the i th row added to it. Then E (c × i + j) A = B where B is obtained from A by replacing the j th row of A with itself added to c times the i th row of A.
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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
Page 57 and 58: 2.4. SUBSPACES AND SPANS 57 Proof:
Page 59 and 60: 2.4. SUBSPACES AND SPANS 59 Proof 3
Page 61 and 62: 2.5. AN APPLICATION TO MATRICES 61
Page 63 and 64: 2.6. MATRICES AND CALCULUS 63 2.6.1
Page 65 and 66: 2.6. MATRICES AND CALCULUS 65 where
Page 67 and 68: 2.6. MATRICES AND CALCULUS 67 There
Page 69 and 70: 2.6. MATRICES AND CALCULUS 69 Using
Page 71 and 72: 2.7. EXERCISES 71 this will suffice
Page 73 and 74: 2.7. EXERCISES 73 22. If A is a lin
Page 75 and 76: 2.7. EXERCISES 75 35. ↑Suppose yo
Page 77 and 78: Determinants 3.1 Basic Techniques A
Page 79 and 80: 3.1. BASIC TECHNIQUES AND PROPERTIE
Page 81 and 82: 3.2. EXERCISES 81 First note det (A
Page 83 and 84: 3.3. THE MATHEMATICAL THEORY OF DET
Page 97 and 98: 3.4. THE CAYLEY HAMILTON THEOREM 97
Page 99 and 100: 3.5. BLOCK MULTIPLICATION OF MATRIC
Page 101 and 102: 3.5. BLOCK MULTIPLICATION OF MATRIC
Page 103 and 104: 3.6. EXERCISES 103 derivatives so i
Page 105 and 106: Row Operations 4.1 Elementary Matri
Page 107: 4.1. ELEMENTARY MATRICES 107 This h
Page 111 and 112: 4.2. THE RANK OF A MATRIX 111 So ho
Page 113 and 114: 4.3. THE ROW REDUCED ECHELON FORM 1
Page 115 and 116: 4.3. THE ROW REDUCED ECHELON FORM 1
Page 117 and 118: 4.5. FREDHOLM ALTERNATIVE 117 Proof
Page 119 and 120: 4.6. EXERCISES 119 5. ↑Consider t
Page 121 and 122: 4.6. EXERCISES 121 18. Suppose A is
Page 123 and 124: Some Factorizations 5.1 LU Factoriz
Page 125 and 126: 5.3. SOLVING LINEAR SYSTEMS USING A
Page 127 and 128: 5.5. JUSTIFICATION FOR THE MULTIPLI
Page 129 and 130: 5.6. EXISTENCE FOR THE PLU FACTORIZ
Page 131 and 132: 5.7. THE QR FACTORIZATION 131 so it
Page 133 and 134: 5.8. EXERCISES 133 Theorem 5.7.5 Le
Page 135 and 136: Linear Programming 6.1 Simple Geome
Page 137 and 138: 6.2. THE SIMPLEX TABLEAU 137 set x
Page 139 and 140: 6.2. THE SIMPLEX TABLEAU 139 lution
Page 141 and 142: 6.3. THE SIMPLEX ALGORITHM 141 Let
Page 143 and 144: 6.3. THE SIMPLEX ALGORITHM 143 the
Page 145 and 146: 6.3. THE SIMPLEX ALGORITHM 145 by 2
Page 147 and 148: 6.3. THE SIMPLEX ALGORITHM 147 I ne
Page 149 and 150: 6.3. THE SIMPLEX ALGORITHM 149 Of c
Page 151 and 152: 6.4. FINDING A BASIC FEASIBLE SOLUT
Page 153 and 154: 6.5. DUALITY 153 Now recall that to
Page 155 and 156: 6.5. DUALITY 155 and there are stil
Page 157 and 158: 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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14.3. THE SPECTRAL RADIUS 349 ⎛ =
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14.4. SERIES AND SEQUENCES OF LINEA
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14.4. SERIES AND SEQUENCES OF LINEA
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14.5. ITERATIVE METHODS FOR LINEAR
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14.6. THEORY OF CONVERGENCE 361 Lem
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14.7. EXERCISES 363 Proof: From Lem
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14.7. EXERCISES 365 11. Suppose ρ
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14.7. EXERCISES 367 Next take e tλ
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14.7. EXERCISES 369 24. Suppose A i
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Numerical Methods For Finding Eigen
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15.1. THE POWER METHOD FOR EIGENVAL
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15.2. THE QR ALGORITHM 387 each a v
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15.2. THE QR ALGORITHM 389 Proof: L
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15.2. THE QR ALGORITHM 391 where
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15.2. THE QR ALGORITHM 393 Corollar
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15.2. THE QR ALGORITHM 395 below th
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15.2. THE QR ALGORITHM 397 where R
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15.2. THE QR ALGORITHM 399 A matrix
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15.3. EXERCISES 401 where A k = Q k
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Positive Matrices Earlier theorems
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405 |μ| ≤λ 0 . But also, the fa
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407 which says x 0 −x 0 v T x 0 =
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409 must have the same argument for
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Functions Of Matrices The existence
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413 There is no convergence problem
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415 This gives us a nice definition
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Applications To Differential Equati
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C.3. LOCAL SOLUTIONS 419 C.3 Local
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C.4. FIRST ORDER LINEAR SYSTEMS 421
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C.5. GEOMETRIC THEORY OF AUTONOMOUS
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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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