Linear Algebra, Theory And Applications, 2012a

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272 LINEAR TRANSFORMATIONS CANONICAL FORMS where the second factor is irreducible over Q. Consider φ 2 (λ) first. Messy computations yield ⎛ ⎞ −16 −16 −16 −16 −32 0 0 0 0 0 φ 2 (A) = ⎜ 0 0 0 0 0 ⎟ ⎝ 0 0 0 0 0 ⎠ 16 16 16 16 32 and so ⎛ ker (φ 2 (A)) = a ⎜ ⎝ −1 1 0 0 0 ⎞ ⎛ ⎟ ⎠ + b ⎜ ⎝ −1 0 1 0 0 ⎞ ⎛ ⎟ ⎠ + c ⎜ ⎝ −1 0 0 1 0 ⎞ ⎛ ⎟ ⎠ + d ⎜ ⎝ Now start with one of these basis vectors and look for an A cycle. Picking the first one, you obtain the cycle ⎛ ⎞ ⎛ ⎞ −1 −15 1 ⎜ 0 ⎟ ⎝ 0 ⎠ , 5 ⎜ 1 ⎟ ⎝ −5 ⎠ 0 7 because the next vector involving A 2 yields a vector which is in the span of the above two. You check this by making the vectors the columns of a matrix and finding the row reduced echelon form. Clearly this cycle does not span ker (φ 2 (A)) , so look for another cycle. Begin with a vector which is not in the span of these two. The last one works well. Thus another A cycle is ⎛ ⎜ ⎝ It follows a basis for ker (φ 2 (A)) is ⎧⎛ ⎞ ⎛ −2 ⎪⎨ 0 ⎜ 0 ⎟ ⎝ 0 ⎠ , ⎜ ⎝ ⎪⎩ 1 −2 0 0 0 1 −16 4 −4 0 8 ⎞ ⎛ ⎟ ⎠ , ⎜ ⎝ ⎞ ⎛ ⎟ ⎠ , ⎜ ⎝ −16 4 −4 0 8 −1 1 0 0 0 ⎞ ⎟ ⎠ ⎞ ⎛ ⎟ ⎠ , ⎜ ⎝ −15 5 1 −5 7 ⎞⎫ ⎪⎬ ⎟ ⎠ ⎪⎭ From the above theory, these vectors are linearly independent. Finally consider a cycle coming from ker (φ 1 (A)). This amounts to nothing more than finding an eigenvector for A corresponding to the eigenvalue 4. An eigenvector is ( −1 0 0 0 1 ) T .Now the desired matrix for the similarity transformation is ⎛ ⎞ −2 −16 −1 −15 −1 0 4 1 5 0 S ≡ ⎜ 0 −4 0 1 0 ⎟ ⎝ 0 0 0 −5 0 ⎠ 1 8 0 7 1 −2 0 0 0 1 ⎞ ⎟ ⎠ .
10.9. EXERCISES 273 Then doing the computations, you get ⎛ S −1 AS = ⎜ ⎝ 0 −32 0 0 0 1 8 0 0 0 0 0 0 −32 0 0 0 1 8 0 0 0 0 0 4 and you see this is in rational canonical form, the two 2×2 blocks being companion matrices for the polynomial λ 2 −8λ+32 and the 1×1 block being a companion matrix for λ−4. Note that you could have written this without finding a similarity transformation to produce it. This follows from the above theory which gave the existence of the rational canonical form. Obviously there is a lot more which could be considered about rational canonical forms. Just begin with a strange field and start investigating what can be said. It is as far as I feel like going on this subject at this time. One can also derive more systematic methods for finding the rational canonical form. The advantage of this is you don’t need to find the eigenvalues in order to compute the rational canonical form and it can often be computed for this reason, unlike the Jordan form. The uniqueness of this rational canonical form can be used to determine whether two matrices consisting of entries in some field are similar. 10.9 Exercises 1. Letting A be a complex n × n matrix, in obtaining the rational canonical form, one obtains the C n as a direct sum of the form span ( β x1 ) ⊕···⊕span ( βxr ) where β x is an ordered cyclic set of vectors, x,Ax, ··· ,A m−1 x such that A m x is in the span of the previous vectors. Now apply the Gram Schmidt process to the ordered basis ( ) β x1 ,β x2 , ··· ,β xr , the vectors in each βxi listed according to increasing power of A, thus obtaining an ordered basis (q 1 , ··· , q n ) . Letting Q be the unitary matrix which has these vectors as columns, show that Q ∗ AQ equals a matrix B which satisfies B ij =0ifi − j ≥ 2. Such a matrix is called an upper Hessenberg matrix and this shows that every n × n matrix is orthogonally similar to an upper Hessenberg matrix. These are zero below the main sub diagonal, like companion matrices discussed above. 2. In the argument for Theorem 10.2.4 it was shown that m (A) φ l (A) v = v whenever v ∈ ker (φ k (A) r k ) . Show that m (A) restricted to ker (φ k (A) r k ) is the inverse of the linear transformation φ l (A) onker(φ k (A) r k ) . 3. Suppose A is a linear transformation and let the characteristic polynomial be det (λI − A) = q∏ φ j (λ) n j where the φ j (λ) are irreducible. Explain using Corollary 8.3.11 why the irreducible factors of the minimal polynomial are φ j (λ) and why the minimal polynomial is of the form q∏ φ j (λ) r j where r j ≤ n j . You can use the Cayley Hamilton theorem if you like. j=1 j=1 ⎞ ⎟ ⎠
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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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Page 251 and 252: 10.3. CYCLIC SETS 251 while ⎛ ⎞
Page 253 and 254: 10.3. CYCLIC SETS 253 Since β x is
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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
Page 269 and 270: 10.8. UNIQUENESS 269 Thus the matri
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Page 281 and 282: 11.3. MARKOV CHAINS 281 After many
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Page 299 and 300: 12.7. EXERCISES 299 4. Let ||x||
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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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13.6. 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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