Linear Algebra, Theory And Applications, 2012a

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496 ANSWERS TO SELECTED EXERCISES G.20 Exercises G.23 Exercises 12.9 1 volume is √ 218 3 0. G.21 Exercises 13.12 13 This is easy because you show it preserves distances. 15 (Ax, x) =(UDU ∗ x, x) =(DU ∗ x,U ∗ x) ≥ δ 2 |U ∗ x| 2 = δ 2 |x| 2 16 0 > ((A + A ∗ ) x, x) =(Ax, x)+(A ∗ x, x) =(Ax, x) +(Ax, x) NowletAx = λx. Then you get 0 >λ|x| 2 + ¯λ |x| 2 =Re(λ) |x| 2 19 If Ax = λx, then you can take the norm of both sides and conclude that |λ| =1. It follows that the eigenvalues of A are e iθ ,e −iθ and another one which has magnitude 1 and is real. This can only be 1 or −1. Since the determinant is given to be 1, it follows that it is 1. Therefore, there exists an eigenvector for the eigenvalue 1. G.22 Exercises 14.7 ⎛ 1 ⎝ 0.09 ⎞ 0.21 ⎠ 0.43 ⎛ ⎞ 4. 237 3 × 10−2 3 ⎝ 7. 627 1 × 10 −2 ⎠ 0.711 86 28 You have H = U ∗ DU where U is unitary and D is a real diagonal matrix. Then you have ⎛ ⎞ ∑ ∞ e iH = U ∗ (iD) n e iλ1 U = U ∗ ⎜ ⎝ . .. ⎟ ⎠ U n! e iλn n=0 and this is clearly unitary because each matrix in the product is. 15.3 1 2 3 4 ⎛ ⎝ 1 2 3 ⎞ 2 2 1.0 ⎠, eigenvectors: 3 1 4 ⎧⎛ ⎞⎫ ⎨ 0.534 91 ⎬ ⎝ 0.390 22 ⎠ ↔ 6. 662, ⎩ ⎭ 0.749 4 ⎧⎛ ⎞⎫ ⎨ 0.130 16 ⎬ ⎝ 0.838 32 ⎠ ↔ 1. 679 0, ⎩ ⎭ −0.529 42 ⎧⎛ ⎞⎫ ⎨ 0.834 83 ⎬ ⎝ −0.380 73 ⎠ ↔−1. 341 ⎩ ⎭ −0.397 63 ⎛ ⎝ 3 2 1.0 ⎞ 2 1 3 ⎠, eigenvectors: 1 3 2 ⎧⎛ ⎞⎫ ⎨ 0.577 35 ⎬ ⎝ 0.577 35 ⎠ ⎩ ⎭ ↔ 6.0, 0.577 35 ⎧⎛ ⎞⎫ ⎨ 0.788 68 ⎬ ⎝ −0.211 32 ⎠ ↔ 1. 732 1, ⎩ ⎭ −0.577 35 ⎧⎛ ⎞⎫ ⎨ 0.211 32 ⎬ ⎝ −0.788 68 ⎠ ↔−1. 732 1 ⎩ ⎭ 0.577 35 ⎛ ⎝ 3 2 1.0 ⎞ 2 5 3 ⎠, eigenvectors: 1 3 2 ⎧⎛ ⎞⎫ ⎨ 0.416 01 ⎬ ⎝ 0.779 18 ⎠ ↔ 7. 873 0, ⎩ ⎭ 0.468 85 ⎧⎛ ⎞⎫ ⎨ 0.904 53 ⎬ ⎝ −0.301 51 ⎠ ⎩ ⎭ ↔ 2.0, −0.301 51 ⎧⎛ ⎞⎫ ⎨ 9. 356 8 × 10−2 ⎬ ⎝ −0.549 52 ⎠ ↔ 0.127 02 ⎩ ⎭ 0.830 22 ⎛ ⎝ 0 2 1.0 ⎞ 2 5 3 ⎠, eigenvectors: 1 3 2 ⎧⎛ ⎞⎫ ⎨ 0.284 33 ⎬ ⎝ 0.819 59 ⎠ ↔ 7. 514 6, ⎩ ⎭ 0.497 43
G.23. EXERCISES 497 ⎧⎛ ⎞⎫ ⎨ 0.209 84 ⎬ ⎝ 0.453 06 ⎠ ↔ 0.189 11, ⎩ ⎭ −0.866 43 ⎧⎛ ⎞⎫ ⎨ 0.935 48 ⎬ ⎝ −0.350 73 ⎠ ↔−0.703 70 ⎩ 4. 316 8 × 10 −2 ⎭ ⎛ 0 2 ⎞ 1.0 5 ⎝ 2 0 3 ⎠, eigenvectors: 1 3 2 ⎧⎛ ⎨ ⎝ 0.379 2 ⎞⎫ ⎬ 0.584 81 ⎠ ↔ 4. 975 4, ⎩ ⎭ 0.717 08 ⎧⎛ ⎞⎫ ⎨ 0.814 41 ⎬ ⎝ 0.156 94 ⎠ ↔−0.300 56, ⎩ ⎭ −0.558 66 ⎧⎛ ⎞⎫ ⎨ 0.439 25 ⎬ ⎝ −0.795 85 ⎠ ↔−2. 674 9 ⎩ ⎭ 0.416 76 6 |7. 333 3 − λ q |≤0.471 41 7 |7 − λ q | =2. 449 5 8 |λ q − 8| ≤3. 266 0 9 −10 ≤ λ ≤ 12 10 x 3 +7x 2 +3x +7.0 = 0, Solution is: ⎧ ⎫ ⎨ [x = −0.145 83 + 1. 011i] , ⎬ [x = −0.145 83 − 1. 011i] , ⎩ ⎭ [x = −6. 708 3] 11 −1. 475 5 + 1. 182 7i, −1. 475 5 − 1. 182 7i, −0.024 44 + 0.528 23i, −0.024 44 − 0.528 23i 12 Let Q T AQ = H where H is upper Hessenberg. Then take the transpose of both sides. This will show that H = H T and so H iszeroonthetopas well.
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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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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
Page 445 and 446: Fields And Field Extensions F.1 The
Page 447 and 448: F.2. THE FUNDAMENTAL THEOREM OF ALG
Page 449 and 450: F.2. THE FUNDAMENTAL THEOREM OF ALG
Page 451 and 452: F.3. TRANSCENDENTAL NUMBERS 451 F.3
Page 453 and 454: F.3. TRANSCENDENTAL NUMBERS 453 It
Page 455 and 456: F.3. TRANSCENDENTAL NUMBERS 455 Not
Page 457 and 458: F.3. TRANSCENDENTAL NUMBERS 457 Thi
Page 459 and 460: F.4. MORE ON ALGEBRAIC FIELD EXTENS
Page 465 and 466: F.5. THE GALOIS GROUP 465 a rationa
Page 467 and 468: F.5. THE GALOIS GROUP 467 Now let
Page 469 and 470: F.6. NORMAL SUBGROUPS 469 If H is a
Page 471 and 472: F.8. CONDITIONS FOR SEPARABILITY 47
Page 473 and 474: F.8. CONDITIONS FOR SEPARABILITY 47
Page 475 and 476: F.9. PERMUTATIONS 475 of f (x) ands
Page 477 and 478: F.9. PERMUTATIONS 477 smallest k
Page 479 and 480: F.10. SOLVABLE GROUPS 479 Then i 1
Page 481 and 482: F.10. SOLVABLE GROUPS 481 Thus ever
Page 483 and 484: F.11. SOLVABILITY BY RADICALS 483 A
Page 485 and 486: Bibliography [1] Apostol T., Calcul
Page 487 and 488: Answers To Selected Exercises G.1 E
Page 489 and 490: G.7. EXERCISES 489 15 Find the matr
Page 491 and 492: G.10. EXERCISES 491 11 It follows f
Page 493 and 494: G.13. EXERCISES 493 19 ⎛ ⎝ 4
Page 495: G.19. EXERCISES 495 8 λ 3 − λ 2
Page 499 and 500: INDEX 499 defective, 162 DeMoivre i
Page 501 and 502: INDEX 501 rotation, 235 linear tran
Page 503: INDEX 503 scalars, 18, 32, 37 Schur
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