Linear Algebra, Theory And Applications, 2012a

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86 DETERMINANTS A permutation can also be considered as a function from the set {1, 2, ··· ,n} to {1, 2, ··· ,n} as follows. Let f (k) =i k . Permutations are of fundamental importance in certain areas of math. For example, it was by considering permutations that Galois was able to give a criterion for solution of polynomial equations by radicals, but this is a different direction than what is being attempted here. In what follows sgn will often be used rather than sgn n because the context supplies the appropriate n. 3.3.2 The Definition Of The Determinant Definition 3.3.4 Let f be a real valued function which has the set of ordered lists of numbers from {1, ··· ,n} as its domain. Define ∑ f (k 1 ···k n ) (k 1 ,··· ,k n ) to be the sum of all the f (k 1 ···k n ) for all possible choices of ordered lists (k 1 , ··· ,k n ) of numbers of {1, ··· ,n} . For example, ∑ f (k 1 ,k 2 )=f (1, 2) + f (2, 1) + f (1, 1) + f (2, 2) . (k 1 ,k 2 ) Definition 3.3.5 Let (a ij )=A denote an n × n matrix. The determinant of A, denoted by det (A) is defined by det (A) ≡ ∑ sgn (k 1 , ··· ,k n ) a 1k1 ···a nkn (k 1,··· ,k n) where the sum is taken over all ordered lists of numbers from {1, ··· ,n}. Note it suffices to take the sum over only those ordered lists in which there are no repeats because if there are, sgn (k 1 , ··· ,k n )=0and so that term contributes 0 to the sum. Let A be an n × n matrix A =(a ij )andlet(r 1 , ··· ,r n ) denote an ordered list of n numbers from {1, ··· ,n}. LetA (r 1 , ··· ,r n ) denote the matrix whose k th row is the r k row of the matrix A. Thus det (A (r 1 , ··· ,r n )) = ∑ sgn (k 1 , ··· ,k n ) a r1k 1 ···a rnk n (3.7) and A (1, ··· ,n)=A. (k 1 ,··· ,k n ) Proposition 3.3.6 Let (r 1 , ··· ,r n ) be an ordered list of numbers from {1, ··· ,n}. Then sgn (r 1 , ··· ,r n )det(A) = ∑ (k 1 ,··· ,k n ) Proof: Let (1, ··· ,n)=(1, ··· ,r,···s, ··· ,n)sor
3.3. THE MATHEMATICAL THEORY OF DETERMINANTS 87 ∑ (k 1 ,··· ,k n ) sgn (k 1 , ··· ,k r , ··· ,k s , ··· ,k n ) a 1k1 ···a rkr ···a sks ···a nkn , and renaming the variables, calling k s ,k r and k r ,k s , this equals = ∑ sgn (k 1 , ··· ,k s , ··· ,k r , ··· ,k n ) a 1k1 ···a rks ···a skr ···a nkn = ∑ (k 1,··· ,k n) Consequently, (k 1 ,··· ,k n ) ⎛ − sgn ⎝k 1 , ··· , These got switched { }} { k r , ··· ,k s , ··· ,k n ⎞ ⎠ a 1k1 ···a skr ···a rks ···a nkn = − det (A (1, ··· ,s,··· ,r,··· ,n)) . (3.11) det (A (1, ··· ,s,··· ,r,··· ,n)) = − det (A (1, ··· ,r,··· ,s,··· ,n)) = − det (A) Now letting A (1, ··· ,s,··· ,r,··· ,n) play the role of A, and continuing in this way, switching pairs of numbers, det (A (r 1 , ··· ,r n )) = (−1) p det (A) where it took p switches to obtain(r 1 , ··· ,r n )from(1, ··· ,n). By Lemma 3.3.1, this implies det (A (r 1 , ··· ,r n )) = (−1) p det (A) =sgn(r 1 , ··· ,r n )det(A) and proves the proposition in the case when there are no repeated numbers in the ordered list, (r 1 , ··· ,r n ). However, if there is a repeat, say the r th row equals the s th row, then the reasoning of (3.10) -(3.11) shows that det(A (r 1 , ··· ,r n )) = 0 and also sgn (r 1 , ··· ,r n )=0 sotheformulaholdsinthiscasealso. Observation 3.3.7 There are n! ordered lists of distinct numbers from {1, ··· ,n} . To see this, consider n slots placed in order. There are n choices for the first slot. For each of these choices, there are n − 1 choices for the second. Thus there are n (n − 1) ways to fill the first two slots. Then for each of these ways there are n−2 choices left for the third slot. Continuing this way, there are n! ordered lists of distinct numbers from {1, ··· ,n} as stated in the observation. 3.3.3 A Symmetric Definition With the above, it is possible to give a more symmetric description of the determinant from which it will follow that det (A) =det ( A T ) . Corollary 3.3.8 The following formula for det (A) is valid. det (A) = 1 n! · ∑ ∑ (r 1 ,··· ,r n ) (k 1 ,··· ,k n ) sgn (r 1 , ··· ,r n ) sgn (k 1 , ··· ,k n ) a r1k 1 ···a rnk n . (3.12) <strong>And</strong> also det ( A T ) =det(A) where A T is the transpose of A. (Recall that for A T = ( a T ij) , a T ij = a ji.)
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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
Page 35 and 36: 1.15. THE INNER PRODUCT IN F N 35 s
Page 37 and 38: Matrices And Linear Transformations
Page 39 and 40: 2.1. MATRICES 39 Definition 2.1.2 M
Page 41 and 42: 2.1. MATRICES 41 is it possible to
Page 43 and 44: 2.1. MATRICES 43 a3× 3 matrix as d
Page 45 and 46: 2.1. MATRICES 45 1 2 3 4 Write the
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Page 49 and 50: 2.1. MATRICES 49 As described above
Page 51 and 52: 2.2. EXERCISES 51 Write the augment
Page 53 and 54: 2.3. LINEAR TRANSFORMATIONS 53 (e)
Page 55 and 56: 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
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Page 77 and 78: Determinants 3.1 Basic Techniques A
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Page 81 and 82: 3.2. EXERCISES 81 First note det (A
Page 83 and 84: 3.3. THE MATHEMATICAL THEORY OF DET
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Page 103 and 104: 3.6. EXERCISES 103 derivatives so i
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Page 111 and 112: 4.2. THE RANK OF A MATRIX 111 So ho
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Page 117 and 118: 4.5. FREDHOLM ALTERNATIVE 117 Proof
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Page 121 and 122: 4.6. EXERCISES 121 18. Suppose A is
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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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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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