Linear Algebra, Theory And Applications, 2012a

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140 LINEAR PROGRAMMING Example 6.2.3 Consider z = x 1 −x 2 subject to the constraints, x 1 +2x 2 ≤ 10,x 1 +2x 2 ≥ 2, and 2x 1 + x 2 ≤ 6,x i ≥ 0. Find a simplex tableau. Adding in slack variables, an augmented matrix which is descriptive of the constraints is ⎛ ⎝ 1 1 2 2 1 0 0 −1 0 0 10 6 ⎞ ⎠ 2 1 0 0 1 6 The obvious solution is not feasible because of that -1 in the fourth column. When you let x 1 ,x 2 =0, you end up having x 4 = −6 which is negative. Consider the second column and select the 2 as a pivot to zero out that which is above and below the 2. ⎛ 0 0 1 1 0 4 ⎞ ⎝ 1 2 1 0 − 1 2 0 3 ⎠ 3 1 2 0 0 2 1 3 This one is good. When you let x 1 = x 4 =0, you find that x 2 =3,x 3 =4,x 5 =3. The obvious solution is now feasible. You can now assemble the simplex tableau. The first step is to include a column and row for z. This yields ⎛ ⎜ ⎝ 0 0 1 1 0 0 4 1 2 1 0 − 1 2 0 0 3 3 1 2 0 0 2 1 0 3 −1 0 1 0 0 1 0 Now you need to get zeros in the right places so the simple columns will be preserved as simple columns in this larger matrix. This means you need to zero out the 1 in the third column on the bottom. A simplex tableau is now ⎛ ⎜ ⎝ 0 0 1 1 0 0 4 1 2 1 0 − 1 2 0 0 3 3 1 2 0 0 2 1 0 3 −1 0 0 −1 0 1 −4 Note it is not the same one obtained earlier. There is no reason a simplex tableau should be unique. In fact, it follows from the above general description that you have one for each basic feasible point of the region determined by the constraints. 6.3 The Simplex Algorithm 6.3.1 Maximums The simplex algorithm takes you from one basic feasible solution to another while maximizing or minimizing the function you are trying to maximize or minimize. <strong>Algebra</strong>ically, it takes you from one simplex tableau to another in which the lower right corner either increases in the case of maximization or decreases in the case of minimization. I will continue writing the simplex tableau in such a way that the simple columns having only one entry nonzero are on the left. As explained above, this amounts to permuting the variables. I will do this because it is possible to describe what is going on without onerous notation. However, in the examples, I won’t worry so much about it. Thus, from a basic feasible solution, a simplex tableau of the following form has been obtained in which the columns for the basic variables, x B are listed first and b ≥ 0. ( ) I F 0 b 0 c 1 z 0 (6.10) ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ .
6.3. THE SIMPLEX ALGORITHM 141 Let x 0 i = b i for i =1, ··· ,m and x 0 i =0fori>m.Then ( x 0 ,z 0) is a solution to the above system and since b ≥ 0, it follows ( x 0 ,z 0) is a basic feasible solution. ( ) F If c i < 0forsomei, and if F ji ≤ 0 so that a whole column of is ≤ 0withthe c bottom entry < 0, then letting x i be the variable corresponding to that column, you could leave all the other entries of x F equal to zero but change x i to be positive. Let the new vector be denoted by x ′ F and letting x′ B = b − F x′ F it follows (x ′ B) k = b k − ∑ j F kj (x F ) j = b k − F ki x i ≥ 0 Now this shows (x ′ B , x′ F ) is feasible whenever x i > 0 and so you could let x i become arbitrarily large and positive and conclude there is no maximum for z because z =(−c i ) x i + z 0 (6.11) If this happens in a simplex tableau, you can say there is no maximum and stop. What if c ≥ 0? Thenz = z 0 − cx F and to satisfy the constraints, you need x F ≥ 0. Therefore, in this case, z 0 is the largest possible value of z and so the maximum has been found. You stop when this occurs. Next I explain what to do if neither of the above stopping conditions hold. ( The only ) case which remains is that some c i < 0andsomeF ji > 0. You pick a column F in in which c c i < 0, usually the one for which c i is the largest in absolute value. You pick F ji > 0 as a pivot element, divide the j th row by F ji andthenusetoobtain zeros above F ji and below F ji , thus obtaining a new simple column. This row operation also makes exactly one of the other simple columns into a nonsimple column. (In terms of variables, it is said that a free variable becomes a basic variable and a basic variable becomes a free variable.) Now permuting the columns and variables, yields ( I F ′ 0 b ′ 0 c ′ 1 z 0′ ) ( where z 0′ ≥ z 0 because z 0′ = z 0 bj − c i and c i < 0. If b ′ ≥ 0, you are in the same position you were at the beginning but now z 0 is larger. Now here is the important thing. You don’t pick just any F ji when you do these row operations. You pick the positive one for which the row operation results in b ′ ≥ 0. Otherwise the obvious basic feasible solution obtained by letting x ′ F = 0 will fail to satisfy the constraint that x ≥ 0. How is this done? You need F ji ) b ′ k ≡ b k − F kib j ≥ 0 (6.12) F ji for each k =1, ··· ,m or equivalently, b k ≥ F kib j . (6.13) F ji Now if F ki ≤ 0 the above holds. Therefore, you only need to check F pi for F pi > 0. The pivot, F ji is the one which makes the quotients of the form b p F pi
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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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Page 89 and 90: 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
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Page 103 and 104: 3.6. EXERCISES 103 derivatives so i
Page 105 and 106: Row Operations 4.1 Elementary Matri
Page 107 and 108: 4.1. ELEMENTARY MATRICES 107 This h
Page 109 and 110: 4.1. ELEMENTARY MATRICES 109 Now co
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
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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
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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: 6.2. THE SIMPLEX TABLEAU 139 lution
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
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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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Page 165 and 166: 7.2. SOME APPLICATIONS OF EIGENVALU
Page 167 and 168: 7.3. EXERCISES 167 Therefore, the e
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Page 171 and 172: 7.3. EXERCISES 171 38. Let M be an
Page 173 and 174: 7.4. SCHUR’S THEOREM 173 7.4 Schu
Page 175 and 176: 7.4. SCHUR’S THEOREM 175 Proof: L
Page 177 and 178: 7.4. SCHUR’S THEOREM 177 and furt
Page 179 and 180: 7.4. SCHUR’S THEOREM 179 Proof: F
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Page 183 and 184: 7.7. SECOND DERIVATIVE TEST 183 Cor
Page 185 and 186: 7.7. SECOND DERIVATIVE TEST 185 The
Page 187 and 188: 7.9. ADVANCED THEOREMS 187 for find
Page 189 and 190: 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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9.4. EIGENVALUES AND EIGENVECTORS O
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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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