Linear Algebra, Theory And Applications, 2012a

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500 INDEX Gauss Jordan method for inverses, 48 Gauss Seidel method, 356 Gelfand, 349 generalized eigenspace, 75 generalized eigenspaces, 248, 258 generalized eigenvectors, 259 Gerschgorin’s theorem, 186 Gram Schmidt procedure, 134, 173, 290 Gram Schmidt process, 289, 290 Gramm Schmidt process, 173 greatest common divisor, 23, 207 characterization, 23 greatest lower bound, 20 Gronwall’s inequality, 368, 422 group definition, 466 group solvable, 480 Hermitian, 177 orthonormal basis eigenvectors, 313 positive definite, 318 real eigenvalues, 179 Hermitian matrix factorization, 286 positive part, 414 positive part, Lipschitz continuous, 414 Hermitian operator, 293 largest, smallest, eigenvalues, 314 spectral representation, 312 Hessian matrix, 184 Hilbert space, 313 Holder’s inequality, 343 homomorphism, 459 Householder reflection, 131 Householder matrix, 130 idempotent, 72, 489 impossibility of solution by radicals, 483 inconsistent, 29 initial value problem existence, 366, 417 global solutions, 421 linear system, 418 local solutions, existence, uniqueness, 420 uniqueness, 368, 417 injective, 12 inner product, 33, 287 inner product space, 287 adjoint operator, 292 parallelogram identity, 289 triangle inequality, 289 integers mod a prime, 223 integral operator valued function, 367 vector valued function, 367 intersection, 11 intervals notation, 11 invariant, 310 subspace, 249 invariant subspaces direct sum, block diagonal matrix, 250 inverses and determinants, 92 invertible, 47 invertible matrix product of elementary matrices, 115 irreducible, 207 relatively prime, 208 isomorphism, 459 extensions, 461 iterative methods alternate proof of convergence, 365 convergence criterion, 360 diagonally dominant, 365 proof of convergence, 363 Jocobi method, 354 Jordan block, 256, 258 Jordan canonical form existence and uniqueness, 259 powers of a matrix, 260 ker, 115 kernel, 55 kernel of a product direct sum decomposition, 246 Krylov sequence, 251 Lagrange form of remainder, 183 Laplace expansion, 91 least squares, 121, 297, 491 least upper bound, 20 Lindemann Weierstrass theorem, 219, 458 linear combination, 39, 56, 88 linear transformation, 53, 225 defined on a basis, 226 dimension of vector space, 226 existence of eigenvector, 241 kernel, 245 matrix, 54 minimal polynomial, 241
INDEX 501 rotation, 235 linear transformations a vector space, 225 composition, matrices, 234 sum, 225, 295 linearly dependent, 56 linearly independent, 56, 200 linearly independent set extend to basis, 204 Lipschitz condition, 417 LU factorization justification for multiplier method, 127 multiplier method, 123 solutions of linear systems, 125 main diagonal, 78 Markov chain, 279, 280 Markov matrix, 275 limit, 278 regular, 278 steady state, 275, 278 mathematical induction, 21 matrices commuting, 309 notation, 38 transpose, 46 matrix, 37 differentiation operator, 228 injective, 61 inverse, 47 left inverse, 93 lower triangular, 78, 94 Markov, 275 non defective, 177 normal, 177 rank and existence of solutions, 116 rank and nullity, 115 right and left inverse, 61 right inverse, 93 right, left inverse, 93 row, column, determinant rank, 94 self adjoint, 170 stochastic, 275 surjective, 61 symmetric, 169 unitary, 173 upper triangular, 78, 94 matrix exponential, 366 matrix multiplication definition, 40 entries of the product, 42 not commutative, 41 properties, 46 vectors, 39 matrix of linear transformation orthonormal bases, 231 migration matrix, 279 minimal polynomial, 75, 240, 248 eigenvalues, eigenvectors, 241 finding it, 263 generalized eigenspaces, 248 minor, 77, 91 mixed partial derivatives, 182 monic, 207 monomorphism, 459 Moore Penrose inverse, 331 least squares, 332 moving coordinate system, 63 acceleration , 66 negative definite, 318 principle minors, 319 Neuman series, 370 nilpotent block diagonal matrix, 256 Jordan form, uniqueness, 257 Jordan normal form, 256 non defective, 266 non solvable group, 481 nonnegative self adjoint square root, 319 norm, 287 strictly convex, 364 uniformly convex, 364 normal, 324 diagonalizable, 178 non defective, 177 normal closure, 464, 471 normal extension, 463 normal subgroup, 469, 480 normed linear space, 287, 337 normed vector space, 287 norms equivalent, 338 null and rank, 302 null space, 55 nullity, 115 one to one, 12 onto, 12 operator norm, 340
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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
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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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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
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Page 493 and 494: G.13. EXERCISES 493 19 ⎛ ⎝ 4
Page 495 and 496: G.19. EXERCISES 495 8 λ 3 − λ 2
Page 497 and 498: G.23. EXERCISES 497 ⎧⎛ ⎞⎫
Page 499: INDEX 499 defective, 162 DeMoivre i
Page 503: INDEX 503 scalars, 18, 32, 37 Schur
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