Linear Algebra, Theory And Applications, 2012a

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180 SPECTRAL THEORY Then = ⎛ ⎜ ⎝ u T i 1 . u T i n U ′T AU ′ = U ′T ( ) Au i1 ··· Au in ⎞ ⎛ ⎟ ⎠ ( ) ⎜ λ i1 u i1 ··· λ in u in = ⎝ ⎞ λ i1 0 . .. ⎟ ⎠ 0 λ in 7.5 Trace <strong>And</strong> Determinant The determinant has already been discussed. It is also clear that if A = S −1 BS so that A, B are similar, then det (A) = det ( S −1) det (S)det(B) =det ( S −1 S ) det (B) = det(I)det(B) =det(B) The trace is defined in the following definition. Definition 7.5.1 Let A be an n × n matrix whose ij th entry is denoted as a ij .Then trace (A) ≡ ∑ i a ii In other words it is the sum of the entries down the main diagonal. With this definition, it is easy to see that if A = S −1 BS, then trace (A) =trace(B) . Here is why. trace (A) ≡ ∑ i = ∑ j,k A ii = ∑ ( S −1 ) B ij jkS ki = ∑ i,j,k j,k B jk δ kj = ∑ B kk =trace(B) . k ∑ ( B jk S ki S −1 ) ij i Alternatively, trace (AB) ≡ ∑ ij A ij B ji =trace(BA) . Therefore, trace ( S −1 AS ) =trace ( ASS −1) =trace(A) . Theorem 7.5.2 Let A be an n×n matrix. Then trace (A) equals the sum of the eigenvalues of A and det (A) equals the product of the eigenvalues of A. This is proved using Schur’s theorem and is in Problem 17 below. Another important property of the trace is in the following theorem. Theorem 7.5.3 Let A be an m × n matrix and let B be an n × m matrix. Then trace (AB) =trace(BA) . Proof: trace (AB) ≡ ∑ i ( ∑ k A ik B ki ) = ∑ k ∑ B ki A ik =trace(BA) i
7.6. QUADRATIC FORMS 181 7.6 Quadratic Forms Definition 7.6.1 A quadratic form in three dimensions is an expression of the form ⎛ ⎞ ( ) x x y z A ⎝ y ⎠ (7.13) z where A is a 3 × 3 symmetric matrix. In higher dimensions the idea is the same except you use a larger symmetric matrix in place of A. In two dimensions A is a 2 × 2 matrix. For example, consider ⎛ ( ) x y z ⎝ 3 −4 1 ⎞ ⎛ −4 0 −4 ⎠ ⎝ x ⎞ y ⎠ (7.14) 1 −4 3 z which equals 3x 2 − 8xy +2xz − 8yz +3z 2 . This is very awkward because of the mixed terms such as −8xy. The idea is to pick different axes such that if x, y, z are taken with respect to these axes, the quadratic form is much simpler. In other words, look for new variables, x ′ ,y ′ , and z ′ and a unitary matrix U such that ⎛ ⎞ ⎛ U ⎝ x′ y ′ ⎠ = ⎝ x ⎞ y ⎠ (7.15) z ′ z and if you write the quadratic form in terms of the primed variables, there will be no mixed terms. Any symmetric real matrix is Hermitian and is therefore normal. From Corollary 7.4.13, it follows there exists a real unitary matrix U, (an orthogonal matrix) such that U T AU = D a diagonal matrix. Thus in the quadratic form, (7.13) ⎛ ( ) x y z A ⎝ x ⎞ ⎛ ⎞ y ⎠ = ( ) x ′ y ′ z ′ U T AU ⎝ x′ y ′ ⎠ z z ′ ⎛ ⎞ = ( ) x ′ y ′ z ′ D ⎝ x′ y ′ ⎠ z ′ and in terms of these new variables, the quadratic form becomes λ 1 (x ′ ) 2 + λ 2 (y ′ ) 2 + λ 3 (z ′ ) 2 where D =diag(λ 1 ,λ 2 ,λ 3 ) . Similar considerations apply equally well in any other dimension. For the given example, ⎛ − 1 ⎞ ⎛ 2√ 2 0 1 √ √ 2√ 2 ⎝ 1 6 6 1 3 6 1 √ 6√ 6 ⎠ ⎝ 3 −4 1 ⎞ √ √ −4 0 −4 ⎠ · 1 3 3 − 1 3 3 1 3 3 1 −4 3 ⎛ ⎞ − √ 1 √6 1 √3 1 ⎛ ⎜ 2 2 ⎝ 0 √6 − √ 1 ⎟ 3 ⎠ = ⎝ 2 0 0 ⎞ 0 −4 0 ⎠ 1√ √6 1 √3 1 0 0 8 2
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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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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
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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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Page 167 and 168: 7.3. EXERCISES 167 Therefore, the e
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Page 177 and 178: 7.4. SCHUR’S THEOREM 177 and furt
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Page 185 and 186: 7.7. SECOND DERIVATIVE TEST 185 The
Page 187 and 188: 7.9. ADVANCED THEOREMS 187 for find
Page 189 and 190: 7.9. ADVANCED THEOREMS 189 Proof: D
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Page 201 and 202: 8.2. SUBSPACES AND BASES 201 8.2.2
Page 203 and 204: 8.2. SUBSPACES AND BASES 203 Defini
Page 205 and 206: 8.3. LOTS OF FIELDS 205 such that 0
Page 207 and 208: 8.3. LOTS OF FIELDS 207 and this is
Page 209 and 210: 8.3. LOTS OF FIELDS 209 Lemma 8.3.9
Page 211 and 212: 8.3. LOTS OF FIELDS 211 Definition
Page 213 and 214: 8.3. LOTS OF FIELDS 213 Then, since
Page 215 and 216: 8.3. LOTS OF FIELDS 215 Proposition
Page 217 and 218: 8.3. LOTS OF FIELDS 217 where deg r
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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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