Linear Algebra, 2020a

More documents

Recommendations

Info

466 Chapter Five. Similarity such as tridiagonal form, where the only nonzero entries are on the diagonal, or just above or below it. Then special case techniques can find the eigenvalues. Once we know the eigenvalues then we can easily compute the eigenvectors of T. These other methods are outside of our scope. A good reference is [Goult, et al.] Exercises 1 Use ten iterations to estimate the largest eigenvalue of these matrices, starting from the vector with components 1 and 2. Compare the answer with the one obtained by solving the characteristic equation. (a) ( 1 5 0 4 ) (b) ( 3 2 −1 0 ) 2 Redo the prior exercise by iterating until |⃗v k | − |⃗v k−1 | has absolute value less than 0.01 At each step, normalize by dividing each vector by its length. How many iterations does it take? Are the answers significantly different? 3 Use ten iterations to estimate the largest eigenvalue of these matrices, starting from the vector with components 1, 2, and 3. Compare the answer with the one obtained by solving the characteristic equation. ⎛ ⎞ ⎛ ⎞ 4 0 1 −1 2 2 (a) ⎝ −2 1 0 −2 0 1 ⎠ (b) ⎝ 2 2 2 −3 −6 −6 4 Redo the prior exercise by iterating until |⃗v k | − |⃗v k−1 | has absolute value less than 0.01. At each step, normalize by dividing each vector by its length. How many iterations does it take? Are the answers significantly different? 5 What happens if c 1 = 0? That is, what happens if the initial vector does not to have any component in the direction of the relevant eigenvector? 6 How can we adapt the method of powers to find the smallest eigenvalue? Computer Code This is the code for the computer algebra system Octave that did the calculation above. (It has been lightly edited to remove blank lines, etc.) >T=[3, 0; 8, -1] T= 3 0 8 -1 >v0=[1; 2] v0= 1 1 >v1=T*v0 v1= 3 7 >v2=T*v1 v2= 9 17 >T9=T**9 T9= 19683 0 ⎠
Topic: Method of Powers 467 39368 -1 >T10=T**10 T10= 59049 0 118096 1 >v9=T9*v0 v9= 19683 39367 >v10=T10*v0 v10= 59049 118096 >norm(v10)/norm(v9) ans=2.9999 Remark. This does not use the full power of Octave; it has built-in functions to automatically apply sophisticated methods to find eigenvalues and eigenvectors.
Page 1 and 2:
LINEAR ALGEBRA Jim Hefferon Fourth
Page 3 and 4:
Preface This book helps students to
Page 5 and 6:
Advice. This book’s emphasis on m
Page 7 and 8:
Contents Chapter One: Linear System
Page 9:
III Laplace’s Formula ...........
Page 12 and 13:
2 Chapter One. Linear Systems must
Page 14 and 15:
4 Chapter One. Linear Systems 1.5 T
Page 16 and 17:
6 Chapter One. Linear Systems 1.9 E
Page 18 and 19:
8 Chapter One. Linear Systems 1.14
Page 20 and 21:
10 Chapter One. Linear Systems ̌ 1
Page 22 and 23:
Page 24 and 25:
14 Chapter One. Linear Systems free
Page 26 and 27:
16 Chapter One. Linear Systems abov
Page 28 and 29:
18 Chapter One. Linear Systems We w
Page 30 and 31:
20 Chapter One. Linear Systems We
Page 32 and 33:
22 Chapter One. Linear Systems reve
Page 34 and 35:
Page 36 and 37:
26 Chapter One. Linear Systems eche
Page 38 and 39:
28 Chapter One. Linear Systems To f
Page 40 and 41:
30 Chapter One. Linear Systems Proo
Page 42 and 43:
32 Chapter One. Linear Systems than
Page 44 and 45:
34 Chapter One. Linear Systems (a)
Page 46 and 47:
36 Chapter One. Linear Systems to f
Page 48 and 49:
38 Chapter One. Linear Systems The
Page 50 and 51:
40 Chapter One. Linear Systems line
Page 52 and 53:
42 Chapter One. Linear Systems ̌ 1
Page 54 and 55:
44 Chapter One. Linear Systems Note
Page 56 and 57:
Page 58 and 59:
Page 60 and 61:
50 Chapter One. Linear Systems III
Page 62 and 63:
52 Chapter One. Linear Systems elem
Page 64 and 65:
54 Chapter One. Linear Systems One
Page 66 and 67:
56 Chapter One. Linear Systems III.
Page 68 and 69:
58 Chapter One. Linear Systems We n
Page 70 and 71:
60 Chapter One. Linear Systems and
Page 72 and 73:
Page 74 and 75:
Page 76 and 77:
66 Chapter One. Linear Systems [0,0
Page 78 and 79:
68 Chapter One. Linear Systems Now
Page 80 and 81:
70 Chapter One. Linear Systems (b)
Page 82 and 83:
Topic Accuracy of Computations Gaus
Page 84 and 85:
74 Chapter One. Linear Systems The
Page 86 and 87:
Topic Analyzing Networks The diagra
Page 88 and 89:
78 Chapter One. Linear Systems and
Page 90 and 91:
80 Chapter One. Linear Systems (c)
Page 93 and 94:
Chapter Two Vector Spaces The first
Page 95 and 96:
Section I. Definition of Vector Spa
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Section II. Linear Independence 109
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Section III. Basis and Dimension 12
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Topic Fields Computations involving
Page 165 and 166:
Topic Crystals Everyone has noticed
Page 167 and 168:
Topic: Crystals 157 is a good basis
Page 169 and 170:
Topic Voting Paradoxes Imagine that
Page 171 and 172:
Topic: Voting Paradoxes 161 subspac
Page 173 and 174:
Topic: Voting Paradoxes 163 between
Page 175 and 176:
Topic Dimensional Analysis “You c
Page 177 and 178:
Topic: Dimensional Analysis 167 for
Page 179 and 180:
Topic: Dimensional Analysis 169 whi
Page 181 and 182:
Topic: Dimensional Analysis 171 (b)
Page 183 and 184:
Chapter Three Maps Between Spaces I
Page 185 and 186:
Section I. Isomorphisms 175 and sca
Page 187 and 188:
Section I. Isomorphisms 177 The fir
Page 189 and 190:
Section I. Isomorphisms 179 picture
Page 191 and 192:
Section I. Isomorphisms 181 (b) f:
Page 193 and 194:
Section I. Isomorphisms 183 (d) Pro
Page 195 and 196:
Section I. Isomorphisms 185 2.3 The
Page 197 and 198:
Section I. Isomorphisms 187 In the
Page 199 and 200:
Section I. Isomorphisms 189 Associa
Page 201 and 202:
Section II. Homomorphisms 191 II Ho
Page 203 and 204:
Section II. Homomorphisms 193 So on
Page 205 and 206:
Section II. Homomorphisms 195 1.12
Page 207 and 208:
Page 209 and 210:
Page 211 and 212:
Section II. Homomorphisms 201 Recal
Page 213 and 214:
Section II. Homomorphisms 203 Now,
Page 215 and 216:
Page 217 and 218:
Section II. Homomorphisms 207 Exerc
Page 219 and 220:
Section II. Homomorphisms 209 (a) h
Page 221 and 222:
Page 223 and 224:
Section III. Computing Linear Maps
Page 225 and 226:
Page 227 and 228:
Page 229 and 230:
Page 231 and 232:
Page 233 and 234:
Page 235 and 236:
Page 237 and 238:
Page 239 and 240:
Page 241 and 242:
Page 243 and 244:
Section IV. Matrix Operations 233 1
Page 245 and 246:
Section IV. Matrix Operations 235 (
Page 247 and 248:
Section IV. Matrix Operations 237 D
Page 249 and 250:
Section IV. Matrix Operations 239 A
Page 251 and 252:
Section IV. Matrix Operations 241 e
Page 253 and 254:
Page 255 and 256:
Section IV. Matrix Operations 245 T
Page 257 and 258:
Section IV. Matrix Operations 247 a
Page 259 and 260:
Page 261 and 262:
Section IV. Matrix Operations 251 a
Page 263 and 264:
Section IV. Matrix Operations 253 R
Page 265 and 266:
Section IV. Matrix Operations 255 W
Page 267 and 268:
Page 269 and 270:
Page 271 and 272:
Page 273 and 274:
Section V. Change of Basis 263 1.2
Page 275 and 276:
Section V. Change of Basis 265 to t
Page 277 and 278:
Section V. Change of Basis 267 ̌ 1
Page 279 and 280:
Section V. Change of Basis 269 for
Page 281 and 282:
Section V. Change of Basis 271 beca
Page 283 and 284:
Section V. Change of Basis 273 Exer
Page 285 and 286:
Section VI. Projection 275 VI Proje
Page 287 and 288:
Section VI. Projection 277 1.4 Exam
Page 289 and 290:
Section VI. Projection 279 (a) ( 1
Page 291 and 292:
Section VI. Projection 281 For the
Page 293 and 294:
Section VI. Projection 283 By the f
Page 295 and 296:
Section VI. Projection 285 (c) Let
Page 297 and 298:
Section VI. Projection 287 We will
Page 299 and 300:
Section VI. Projection 289 of the v
Page 301 and 302:
Section VI. Projection 291 is perpe
Page 303 and 304:
Section VI. Projection 293 3.19 Wha
Page 305 and 306:
Topic Line of Best Fit This Topic r
Page 307 and 308:
Topic: Line of Best Fit 297 the sam
Page 309 and 310:
Topic: Line of Best Fit 299 4 Find
Page 311 and 312:
Topic Geometry of Linear Maps These
Page 313 and 314:
Topic: Geometry of Linear Maps 303
Page 315 and 316:
Page 317 and 318:
Page 319 and 320:
Topic: Magic Squares 309 One interp
Page 321 and 322:
Topic: Magic Squares 311 1 2 3 4 5
Page 323 and 324:
Topic Markov Chains Here is a simpl
Page 325 and 326:
Topic: Markov Chains 315 a middle c
Page 327 and 328:
Topic: Markov Chains 317 3 [Kelton]
Page 329 and 330:
Topic Orthonormal Matrices In The E
Page 331 and 332:
Topic: Orthonormal Matrices 321 (we
Page 333 and 334:
Topic: Orthonormal Matrices 323 ( a
Page 335 and 336:
Chapter Four Determinants In the fi
Page 337 and 338:
Section I. Definition 327 A good st
Page 339 and 340:
Section I. Definition 329 the opera
Page 341 and 342:
Section I. Definition 331 is the ve
Page 343 and 344:
Section I. Definition 333 A singula
Page 345 and 346:
Section I. Definition 335 could pos
Page 347 and 348:
Section I. Definition 337 I.3 The P
Page 349 and 350:
Section I. Definition 339 Now this
Page 351 and 352:
Section I. Definition 341 matrix, k
Page 353 and 354:
Section I. Definition 343 Computing
Page 355 and 356:
Section I. Definition 345 3.34 A ma
Page 357 and 358:
Section I. Definition 347 could be
Page 359 and 360:
Section I. Definition 349 Thus, in
Page 361 and 362:
Section I. Definition 351 That is,
Page 363 and 364:
Section I. Definition 353 in the to
Page 365 and 366:
Section II. Geometry of Determinant
Page 367 and 368:
Page 369 and 370:
Page 371 and 372:
Page 373 and 374:
Section III. Laplace’s Formula 36
Page 375 and 376:
Page 377 and 378:
Page 379 and 380:
Topic Cramer’s Rule A linear syst
Page 381 and 382:
Topic: Cramer’s Rule 371 x − y
Page 383 and 384:
Topic: Speed of Calculating Determi
Page 385 and 386:
Topic: Speed of Calculating Determi
Page 387 and 388:
Topic: Chiò’s Method 377 This re
Page 389 and 390:
Topic: Chiò’s Method 379 (a) 2 1
Page 391 and 392:
Topic: Projective Geometry 381 This
Page 393 and 394:
Topic: Projective Geometry 383 Ther
Page 395 and 396:
Topic: Projective Geometry 385 (We
Page 397 and 398:
Topic: Projective Geometry 387 the
Page 399 and 400:
Topic: Projective Geometry 389 The
Page 401 and 402:
Topic: Projective Geometry 391 Exer
Page 403 and 404:
Topic: Computer Graphics 393 In thi
Page 405 and 406:
Topic: Computer Graphics 395 In thi
Page 407 and 408:
Chapter Five Similarity We have sho
Page 409 and 410:
Section I. Complex Vector Spaces 39
Page 411 and 412:
Section I. Complex Vector Spaces 40
Page 413 and 414:
Section II. Similarity 403 represen
Page 415 and 416:
Section II. Similarity 405 A common
Page 417 and 418:
Section II. Similarity 407 ̌ 1.13
Page 419 and 420:
Section II. Similarity 409 2.4 Lemm
Page 421 and 422:
Section II. Similarity 411 Exercise
Page 423 and 424:
Section II. Similarity 413 where x
Page 425 and 426: Section II. Similarity 415 We next
Page 427 and 428: Section II. Similarity 417 3.12 Def
Page 429 and 430: Section II. Similarity 419 3.18 The
Page 431 and 432: Section II. Similarity 421 a criter
Page 433 and 434: Section II. Similarity 423 3.46 Dia
Page 435 and 436: Section III. Nilpotence 425 has thi
Page 437 and 438: Section III. Nilpotence 427 1.7 Exa
Page 439 and 440: Section III. Nilpotence 429 2.2 Cor
Page 441 and 442: Section III. Nilpotence 431 2.9 Exa
Page 443 and 444: Section III. Nilpotence 433 But, th
Page 445 and 446: Section III. Nilpotence 435 (In the
Page 447 and 448: Section III. Nilpotence 437 2.19 Ex
Page 449 and 450: Section III. Nilpotence 439 ̌ 2.24
Page 451 and 452: Section IV. Jordan Form 441 The pol
Page 453 and 454: Section IV. Jordan Form 443 Proof W
Page 455 and 456: Section IV. Jordan Form 445 We refe
Page 457 and 458: Section IV. Jordan Form 447 (a) (e)
Page 459 and 460: Section IV. Jordan Form 449 Convers
Page 461 and 462: Section IV. Jordan Form 451 and to
Page 463 and 464: Section IV. Jordan Form 453 The dia
Page 465 and 466: Section IV. Jordan Form 455 The rig
Page 467 and 468: Section IV. Jordan Form 457 2.12 Ex
Page 469 and 470: Section IV. Jordan Form 459 So the
Page 471 and 472: Section IV. Jordan Form 461 Therefo
Page 473 and 474: Section IV. Jordan Form 463 2.38 In
Page 475: Topic: Method of Powers 465 ⃗v T
Page 479 and 480: Topic: Stable Populations 469 Now i
Page 481 and 482: Topic: Page Ranking 471 importance
Page 483 and 484: Topic: Page Ranking 473 Exercises 1
Page 485 and 486: Topic: Linear Recurrences 475 Write
Page 487 and 488: Topic: Linear Recurrences 477 We ca
Page 489 and 490: Topic: Linear Recurrences 479 place
Page 491 and 492: Topic: Linear Recurrences 481 After
Page 493 and 494: Topic: Coupled Oscillators 483 We s
Page 495 and 496: Topic: Coupled Oscillators 485 we a
Page 497 and 498: Appendix Mathematics is made of arg
Page 499 and 500: A-3 hypothesis. This argument is wr
Page 501 and 502: A-5 Another example is this proof t
Page 503 and 504: A-7 A collection that is like a set
Page 505 and 506: A-9 A function has a left inverse i
Page 507: A-11 related to 102. Verifying the
Page 510 and 511: [Am. Math. Mon., Dec. 1966] Hans Li
Page 512 and 513: [Gardner, 1970] Martin Gardner, Mat
Page 514 and 515: [Online Encyclopedia of Integer Seq
Page 516 and 517: Index accuracy of Gauss’s Method,
Page 518 and 519: elementary reduction operations, 5
Page 520 and 521: linear independence multiset, 113 l
Page 522 and 523: in projective plane, 383, 392 polyn
Page 524 and 525: summation notation, for permutation
show all

Linear Algebra, 2020a

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?