Linear Algebra, Theory And Applications, 2012a

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112 ROW OPERATIONS At this point it is clear the rank is 2. This is because every column is in the span of the first two and these first two columns are linearly independent. Example 4.2.6 Find the rank of the following matrix and identify columns whose linear combinations yield all the other columns. ⎛ ⎝ 1 2 1 3 2 ⎞ 1 2 6 0 2 ⎠ (4.2) 3 6 8 6 6 Take (−1) times the first row and add to the second and then take (−3) times the first row and add to the last row. This yields ⎛ ⎝ 1 2 1 3 2 ⎞ 0 0 5 −3 0 ⎠ 0 0 5 −3 0 Now multiply the second row by 1/5 andadd5timesittothelastrow. ⎛ ⎝ 1 0 2 0 1 1 3 −3/5 2 0 ⎞ ⎠ 0 0 0 0 0 Add (−1) times the second row to the first. ⎛ ⎝ 1 2 0 ⎞ 18 5 2 0 0 1 −3/5 0 ⎠ (4.3) 0 0 0 0 0 It is now clear the rank of this matrix is 2 because the first and third columns form a basis for the column space. The matrix (4.3) is the row reduced echelon form for the matrix (4.2). 4.3 The Row Reduced Echelon Form The following definition is for the row reduced echelon form of a matrix. Definition 4.3.1 Let e i denote the column vector which has all zero entries except for the i th slot which is one. An m×n matrix is said to be in row reduced echelon form if, in viewing successive columns from left to right, the first nonzero column encountered is e 1 and if you have encountered e 1 , e 2 , ··· , e k , thenextcolumniseithere k+1 or is a linear combination of the vectors, e 1 , e 2 , ··· , e k . For example, here are some matrices which are in row reduced echelon form. ⎛ ⎝ 0 1 3 0 3 ⎞ ⎛ 0 0 0 1 5 ⎠ , ⎝ 1 0 3 −11 0 ⎞ 0 1 4 4 0 ⎠ . 0 0 0 0 0 0 0 0 0 1 Theorem 4.3.2 Let A be an m × n matrix. determined by a simple process. Then A has a row reduced echelon form
4.3. THE ROW REDUCED ECHELON FORM 113 Proof: Viewing the columns of A from left to right take the first nonzero column. Pick a nonzero entry in this column and switch the row containing this entry with the top row of A. Now divide this new top row by the value of this nonzero entry to get a 1 in this position and then use row operations to make all entries below this entry equal to zero. Thus the first nonzero column is now e 1 . Denote the resulting matrix by A 1 . Consider the submatrix of A 1 to the right of this column and below the first row. Do exactly the same thing for it that was done for A. Thistimethee 1 will refer to F m−1 . Use this 1 and row operations to zero out every entry above it in the rows of A 1 . Call the resulting matrix A 2 . Thus A 2 satisfies the conditions of the above definition up to the column just encountered. Continue this way till every column has been dealt with and the result must be in row reduced echelon form. The following diagram illustrates the above procedure. Say the matrix looked something like the following. ⎛ ⎞ 0 ∗ ∗ ∗ ∗ ∗ ∗ 0 ∗ ∗ ∗ ∗ ∗ ∗ ⎜ ⎟ ⎝ . . . . . . . ⎠ 0 ∗ ∗ ∗ ∗ ∗ ∗ First step would yield something like ⎛ ⎞ 0 1 ∗ ∗ ∗ ∗ ∗ 0 0 ∗ ∗ ∗ ∗ ∗ ⎜ ⎟ ⎝ . . . . . . . ⎠ 0 0 ∗ ∗ ∗ ∗ ∗ For the second step you look at the lower right corner as described, ⎛ ⎞ ∗ ∗ ∗ ∗ ∗ ⎜ ⎟ ⎝ . . . . . ⎠ ∗ ∗ ∗ ∗ ∗ and if the first column consists of all zeros but the next one is not all zeros, you would get something like this. ⎛ ⎞ 0 1 ∗ ∗ ∗ ⎜ ⎟ ⎝ . . . . . ⎠ 0 0 ∗ ∗ ∗ Thus, after zeroing out the term in the top row above the 1, you get the following for the next step in the computation of the row reduced echelon form for the original matrix. ⎛ ⎞ 0 1 ∗ 0 ∗ ∗ ∗ 0 0 0 1 ∗ ∗ ∗ ⎜ ⎝ . . . . . . ⎟ . ⎠ . 0 0 0 0 ∗ ∗ ∗ Next you look at the lower right matrix below the top two rows and to the right of the first four columns and repeat the process. Definition 4.3.3 The first pivot column of A is the first nonzero column of A. The next pivot column is the first column after this which is not a linear combination of the columns to its left. The third pivot column is the next column after this which is not a linear combination
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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
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 and 108: 4.1. ELEMENTARY MATRICES 107 This h
Page 109 and 110: 4.1. ELEMENTARY MATRICES 109 Now co
Page 111: 4.2. THE RANK OF A MATRIX 111 So ho
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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
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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9.4. 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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