Linear Algebra, Theory And Applications, 2012a

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280 MARKOV CHAINS AND MIGRATION PROCESSES is in state i. The probability that X n+1 is in state j given that X n is in state i is called a one step transition probability. When this probability does not depend on n it is called stationary and this is the case of interest here. Since this probability does not depend on n it can be denoted by p ij . Here is a simple example called a random walk. Example 11.3.1 Let there be n points, x i , and consider a process of something moving randomly from one point to another. Suppose X n is a sequence of random variables which has values {1, 2, ··· ,n} where X n = i indicates the process has arrived at the i th point. Let p ij be the probability that X n+1 has the value j given that X n has the value i. SinceX n+1 must have some value, it must be the case that ∑ j a ij =1. Note this says that the sum over a row equals 1 and so the situation is a little different than the above in which the sum was over a column. As an example, let x 1 ,x 2 ,x 3 ,x 4 be four points taken in order on R and suppose x 1 and x 4 are absorbing. This means that p 4k =0forallk ≠ 4 and p 1k =0forallk ≠1. Otherwise, you can move either to the left or to the right with probability 1 2 . The Markov matrix associated with this situation is ⎛ ⎜ ⎝ 1 0 0 0 .5 0 .5 0 0 .5 0 .5 0 0 0 1 Definition 11.3.2 Let the stationary transition probabilities, p ij be defined above. The resulting matrix having p ij as its ij th entry is called the matrix of transition probabilities. The sequence of random variables for which these p ij are the transition probabilities is called a Markov chain. The matrix of transition probabilities is called a stochastic matrix. The next proposition is fundamental and shows the significance of the powers of the matrix of transition probabilities. Proposition 11.3.3 Let p n ij denote the probability that X n is in state j given that X 0 was in state i. Then p n ij is the ijth entry of the matrix P n where P =(p ij ) . Proof: This is clearly true if n = 1 and follows from the definition of the p ij . Suppose true for n. Then the probability that X n+1 is at j given that X 0 was at i equals ∑ k pn ik p kj because X n must have some value, k, and so this represents all possible ways to go from i to j. You can go from i to1inn steps with probability p i1 and then from 1 to j in one step with probability p 1j and so the probability of this is p n i1 p 1j but you can also go from i to 2 andthenfrom2toj and from i to 3 and then from 3 to j etc. Thus the sum of these is just what is given and represents the probability of X n+1 having the value j given X 0 has the value i. In the above random walk example, lets take a power of the transition probability matrix to determine what happens. Rounding off to two decimal places, ⎛ ⎜ ⎝ 1 0 0 0 .5 0 .5 0 0 .5 0 .5 0 0 0 1 ⎞ ⎟ ⎠ 20 = ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ . 1 0 0 0 . 67 9. 5 × 10 −7 0 . 33 . 33 0 9. 5 × 10 −7 . 67 0 0 0 1 Thus p 21 is about 2/3 while p 32 is about 1/3 and terms like p 22 are very small. You see this seems to be converging to the matrix ⎛ ⎞ 1 0 0 0 ⎜ 2 1 ⎜⎝ 3 0 0 3 ⎟ 1 2 3 0 0 ⎠ . 3 0 0 0 1 ⎞ ⎟ ⎠ .
11.3. MARKOV CHAINS 281 After many iterations of the process, if you start at 2 you will end up at 1 with probability 2/3 and at 4 with probability 1/3. This makes good intuitive sense because it is twice as far from 2 to 4 as it is from 2 to 1. Theorem 11.3.4 The eigenvalues of ⎛ ⎞ 0 p 0 ··· 0 q 0 p ··· 0 . 0 q 0 .. . ⎜ . ⎝ . 0 .. . .. p ⎟ ⎠ 0 . 0 q 0 have absolute value less than 1. Here p + q =1and both p, q > 0. Proof: By Gerschgorin’s theorem, if λ is an eigenvalue, then |λ| ≤1. Now suppose v is an eigenvector for λ. Then ⎛ ⎞ ⎛ ⎞ pv 2 v 1 qv 1 + pv 3 v 2 Av = ⎜ . = λ ⎟ ⎜ . ⎟ ⎝ qv n−2 + pv n ⎠ ⎝ v n−1 ⎠ qv n−1 v n Suppose |λ| =1. Then the top row shows p |v 2 | = |v 1 | so |v 1 | < |v 2 | . Suppose |v 1 | < |v 2 | < ···< |v k | for some k
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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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Page 257 and 258: 10.5. THE JORDAN CANONICAL FORM 257
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Page 267 and 268: 10.7. THE RATIONAL CANONICAL FORM 2
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Page 303 and 304: 12.8. THE DETERMINANT AND VOLUME 30
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Page 307 and 308: Self Adjoint Operators 13.1 Simulta
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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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