Linear Algebra, Theory And Applications, 2012a

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164 SPECTRAL THEORY There is nothing new about finding the eigenvectors for λ = 1 so consider the eigenvalue λ =2+i. You need to solve ⎛ ⎛ ⎝(2 + i) ⎝ 1 0 0 ⎞ ⎛ 0 1 0 ⎠ − ⎝ 1 0 0 ⎞⎞ ⎛ 0 2 −1 ⎠⎠ ⎝ x ⎞ ⎛ y ⎠ = ⎝ 0 ⎞ 0 ⎠ 0 0 1 0 1 2 z 0 In other words, you must consider the augmented matrix ⎛ ⎝ 1+i 0 0 i 0 1 0 0 ⎞ ⎠ 0 −1 i 0 for the solution. Divide the top row by (1 + i) and then take −i times the second row and add to the bottom. This yields ⎛ ⎝ 1 0 0 i 0 1 0 0 ⎞ ⎠ 0 0 0 0 Now multiply the second row by −i to obtain ⎛ ⎝ 1 0 0 0 0 1 −i 0 0 0 0 0 ⎞ ⎠ Therefore, the eigenvectors are of the form ⎛ ⎞ z ⎝ 0 i 1 ⎠ . You should find the eigenvectors for λ =2− i. These are ⎛ z ⎝ 0 ⎞ −i ⎠ . 1 As usual, if you want to get it right you had better check it. ⎛ ⎝ 1 0 0 ⎞ ⎛ 0 2 −1 ⎠ ⎝ 0 ⎞ ⎛ ⎞ ⎛ 0 −i ⎠ = ⎝ −1 − 2i ⎠ =(2− i) ⎝ 0 1 2 1 2 − i so it worked. 0 −i 1 ⎞ ⎠ 7.2 Some <strong>Applications</strong> Of Eigenvalues <strong>And</strong> Eigenvectors Recall that n × n matrices can be considered as linear transformations. If F is a 3 × 3real matrix having positive determinant, it can be shown that F = RU where R is a rotation matrix and U is a symmetric real matrix having positive eigenvalues. An application of this wonderful result, known to mathematicians as the right polar decomposition, is to continuum mechanics where a chunk of material is identified with a set of points in three dimensional space.
7.2. SOME APPLICATIONS OF EIGENVALUES AND EIGENVECTORS 165 The linear transformation, F in this context is called the deformation gradient and it describes the local deformation of the material. Thus it is possible to consider this deformation in terms of two processes, one which distorts the material and the other which just rotates it. It is the matrix U which is responsible for stretching and compressing. This is why in continuum mechanics, the stress is often taken to depend on U which is known in this context as the right Cauchy Green strain tensor. This process of writing a matrix as a product of two such matrices, one of which preserves distance and the other which distorts is also important in applications to geometric measure theory an interesting field of study in mathematics and to the study of quadratic forms which occur in many applications such as statistics. Here I am emphasizing the application to mechanics in which the eigenvectors of U determine the principle directions, those directions in which the material is stretched or compressed to the maximum extent. Example 7.2.1 Find the principle directions determined by the matrix ⎛ ⎞ 29 6 6 11 11 11 ⎝ 6 41 19 ⎠ The eigenvalues are 3, 1, and 1 2 . 11 6 11 44 19 44 It is nice to be given the eigenvalues. The largest eigenvalue is 3 which means that in the direction determined by the eigenvector associated with 3 the stretch is three times as large. The smallest eigenvalue is 1/2 and so in the direction determined by the eigenvector for 1/2 the material is compressed, becoming locally half as long. It remains to find these directions. First consider the eigenvector for 3. It is necessary to solve ⎛ ⎛ ⎝3 ⎝ 1 0 0 ⎞ ⎛ 0 1 0 ⎠ − ⎝ 0 0 1 29 11 6 11 6 11 6 11 41 44 19 44 44 41 44 6 11 19 44 41 44 ⎞⎞ ⎛ ⎠⎠ Thus the augmented matrix for this system of equations is ⎛ ⎞ 4 11 − 6 11 − 6 11 0 ⎝ − 6 91 11 44 − 19 44 0 ⎠ − 6 11 − 19 91 44 44 0 The row reduced echelon form is ⎛ ⎝ 1 0 0 1 −3 −1 0 0 ⎞ ⎠ 0 0 0 0 ⎝ x y z ⎞ ⎛ ⎠ = ⎝ 0 ⎞ 0 ⎠ 0 and so the principle direction for the eigenvalue 3 in which the material is stretched to the maximum extent is ⎛ ⎝ ⎞ 3 1 ⎠ . 1 A direction vector in this direction is ⎛ 3/ √ ⎞ 11 ⎝ 1/ √ 11 1/ √ ⎠ . 11
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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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Page 117 and 118: 4.5. FREDHOLM ALTERNATIVE 117 Proof
Page 119 and 120: 4.6. EXERCISES 119 5. ↑Consider t
Page 121 and 122: 4.6. EXERCISES 121 18. Suppose A is
Page 123 and 124: Some Factorizations 5.1 LU Factoriz
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Page 127 and 128: 5.5. JUSTIFICATION FOR THE MULTIPLI
Page 129 and 130: 5.6. EXISTENCE FOR THE PLU FACTORIZ
Page 131 and 132: 5.7. THE QR FACTORIZATION 131 so it
Page 133 and 134: 5.8. EXERCISES 133 Theorem 5.7.5 Le
Page 135 and 136: Linear Programming 6.1 Simple Geome
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 153 and 154: 6.5. DUALITY 153 Now recall that to
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 173 and 174: 7.4. SCHUR’S THEOREM 173 7.4 Schu
Page 175 and 176: 7.4. SCHUR’S THEOREM 175 Proof: L
Page 177 and 178: 7.4. SCHUR’S THEOREM 177 and furt
Page 179 and 180: 7.4. SCHUR’S THEOREM 179 Proof: F
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Page 183 and 184: 7.7. SECOND DERIVATIVE TEST 183 Cor
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 193 and 194: 7.10. EXERCISES 193 27. Let f (x, y
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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
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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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9.4. 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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