Linear Algebra, 2020a

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472 Chapter Five. Similarity The eigenvector that Sage gives associated with the eigenvalue λ = 1 is this. ⎛ ⎞ 1 2 ⎜ ⎟ ⎝9/4⎠ 1 Of course, there are many vectors in that eigenspace. To get a page rank number we normalize to length one. sage: v=vector([1, 2, 9/4, 1]) sage: v/v.norm() (4/177*sqrt(177), 8/177*sqrt(177), 3/59*sqrt(177), 4/177*sqrt(177)) sage: w=v/v.norm() sage: w.n() (0.300658411201132, 0.601316822402263, 0.676481425202546, 0.300658411201132) So we rank the first and fourth pages as of equal importance. We rank the second and third pages as much more important than those, and about equal in importance as each other. We’ll add one more refinement. We will allow the surfer to pick a new page at random even if they are not on a dangling page. In this equation it happens with probability 1 − α. ⎛ ⎞ ⎛ ⎞ 0 0 1/3 1/4 1/4 1/4 1/4 1/4 1 0 1/3 1/4 G = α · ⎜ ⎟ ⎝0 1 0 1/4⎠ +(1 − α) · 1/4 1/4 1/4 1/4 ⎜ ⎟ ⎝1/4 1/4 1/4 1/4⎠ 0 0 1/3 1/4 1/4 1/4 1/4 1/4 This is the Google matrix . In practice α is typically between 0.85 and 0.99. Here are the ranks for the four pages with various α’s. α 0.85 0.90 0.95 0.99 p 1 0.325 0.317 0.309 0.302 p 2 0.602 0.602 0.602 0.601 p 3 0.652 0.661 0.669 0.675 p 4 0.325 0.317 0.309 0.302 The details of the algorithms used by commercial search engines are secret, have many refinements, and also change frequently. But the inventors of Google were gracious enough to outline the basis for their work in [Brin & Page]. A more current source is [Wikipedia, Google Page Rank]. Two additional excellent expositions are [Wills] and [Austin].
Topic: Page Ranking 473 Exercises 1 A square matrix is stochastic if the sum of the entries in each column is one. The Google matrix is computed by taking a combination G = α ∗ H +(1 − α) ∗ S of two stochastic matrices. Show that G must be stochastic. 2 For this web of pages, the importance of each page should be equal. Verify it for α = 0.85. p 1 p 2 p 4 p 3 3 [Bryan & Leise] Give the importance ranking for this web of pages. p 1 p 2 p 4 p 3 (a) Use α = 0.85. (b) Use α = 0.95. (c) Observe that while p 3 is linked-to from all other pages, and therefore seems important, it is not the highest ranked page. What is the highest ranked page? Explain.
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LINEAR ALGEBRA Jim Hefferon Fourth
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Preface This book helps students to
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Advice. This book’s emphasis on m
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Contents Chapter One: Linear System
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III Laplace’s Formula ...........
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2 Chapter One. Linear Systems must
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4 Chapter One. Linear Systems 1.5 T
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6 Chapter One. Linear Systems 1.9 E
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8 Chapter One. Linear Systems 1.14
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10 Chapter One. Linear Systems ̌ 1
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14 Chapter One. Linear Systems free
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16 Chapter One. Linear Systems abov
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18 Chapter One. Linear Systems We w
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20 Chapter One. Linear Systems We
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22 Chapter One. Linear Systems reve
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26 Chapter One. Linear Systems eche
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28 Chapter One. Linear Systems To f
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30 Chapter One. Linear Systems Proo
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32 Chapter One. Linear Systems than
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34 Chapter One. Linear Systems (a)
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36 Chapter One. Linear Systems to f
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38 Chapter One. Linear Systems The
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40 Chapter One. Linear Systems line
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42 Chapter One. Linear Systems ̌ 1
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44 Chapter One. Linear Systems Note
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50 Chapter One. Linear Systems III
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52 Chapter One. Linear Systems elem
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54 Chapter One. Linear Systems One
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56 Chapter One. Linear Systems III.
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58 Chapter One. Linear Systems We n
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60 Chapter One. Linear Systems and
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66 Chapter One. Linear Systems [0,0
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68 Chapter One. Linear Systems Now
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70 Chapter One. Linear Systems (b)
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Topic Accuracy of Computations Gaus
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74 Chapter One. Linear Systems The
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Topic Analyzing Networks The diagra
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78 Chapter One. Linear Systems and
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80 Chapter One. Linear Systems (c)
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Chapter Two Vector Spaces The first
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Section I. Definition of Vector Spa
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Section II. Linear Independence 109
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Section III. Basis and Dimension 12
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Topic Fields Computations involving
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Topic Crystals Everyone has noticed
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Topic: Crystals 157 is a good basis
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Topic Voting Paradoxes Imagine that
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Topic: Voting Paradoxes 161 subspac
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Topic: Voting Paradoxes 163 between
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Topic Dimensional Analysis “You c
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Topic: Dimensional Analysis 167 for
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Topic: Dimensional Analysis 169 whi
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Topic: Dimensional Analysis 171 (b)
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Chapter Three Maps Between Spaces I
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Section I. Isomorphisms 175 and sca
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Section I. Isomorphisms 177 The fir
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Section I. Isomorphisms 179 picture
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Section I. Isomorphisms 181 (b) f:
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Section I. Isomorphisms 183 (d) Pro
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Section I. Isomorphisms 185 2.3 The
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Section I. Isomorphisms 187 In the
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Section I. Isomorphisms 189 Associa
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Section II. Homomorphisms 191 II Ho
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Section II. Homomorphisms 193 So on
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Section II. Homomorphisms 195 1.12
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Section II. Homomorphisms 201 Recal
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Section II. Homomorphisms 203 Now,
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Section II. Homomorphisms 207 Exerc
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Section II. Homomorphisms 209 (a) h
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Section III. Computing Linear Maps
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Section IV. Matrix Operations 233 1
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Section IV. Matrix Operations 235 (
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Section IV. Matrix Operations 237 D
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Section IV. Matrix Operations 239 A
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Section IV. Matrix Operations 241 e
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Section IV. Matrix Operations 245 T
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Section IV. Matrix Operations 247 a
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Section IV. Matrix Operations 251 a
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Section IV. Matrix Operations 253 R
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Section IV. Matrix Operations 255 W
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Section V. Change of Basis 263 1.2
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Section V. Change of Basis 265 to t
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Section V. Change of Basis 267 ̌ 1
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Section V. Change of Basis 269 for
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Section V. Change of Basis 271 beca
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Section V. Change of Basis 273 Exer
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Section VI. Projection 275 VI Proje
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Section VI. Projection 277 1.4 Exam
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Section VI. Projection 279 (a) ( 1
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Section VI. Projection 281 For the
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Section VI. Projection 283 By the f
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Section VI. Projection 285 (c) Let
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Section VI. Projection 287 We will
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Section VI. Projection 289 of the v
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Section VI. Projection 291 is perpe
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Section VI. Projection 293 3.19 Wha
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Topic Line of Best Fit This Topic r
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Topic: Line of Best Fit 297 the sam
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Topic: Line of Best Fit 299 4 Find
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Topic Geometry of Linear Maps These
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Topic: Geometry of Linear Maps 303
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Topic: Magic Squares 309 One interp
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Topic: Magic Squares 311 1 2 3 4 5
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Topic Markov Chains Here is a simpl
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Topic: Markov Chains 315 a middle c
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Topic: Markov Chains 317 3 [Kelton]
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Topic Orthonormal Matrices In The E
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Topic: Orthonormal Matrices 321 (we
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Topic: Orthonormal Matrices 323 ( a
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Chapter Four Determinants In the fi
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Section I. Definition 327 A good st
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Section I. Definition 329 the opera
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Section I. Definition 331 is the ve
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Section I. Definition 333 A singula
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Section I. Definition 335 could pos
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Section I. Definition 337 I.3 The P
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Section I. Definition 339 Now this
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Section I. Definition 341 matrix, k
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Section I. Definition 343 Computing
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Section I. Definition 345 3.34 A ma
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Section I. Definition 347 could be
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Section I. Definition 349 Thus, in
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Section I. Definition 351 That is,
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Section I. Definition 353 in the to
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Section II. Geometry of Determinant
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Section III. Laplace’s Formula 36
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Topic Cramer’s Rule A linear syst
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Topic: Cramer’s Rule 371 x − y
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Topic: Speed of Calculating Determi
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Topic: Speed of Calculating Determi
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Topic: Chiò’s Method 377 This re
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Topic: Chiò’s Method 379 (a) 2 1
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Topic: Projective Geometry 381 This
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Topic: Projective Geometry 383 Ther
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Topic: Projective Geometry 385 (We
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Topic: Projective Geometry 387 the
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Topic: Projective Geometry 389 The
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Topic: Projective Geometry 391 Exer
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Topic: Computer Graphics 393 In thi
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Topic: Computer Graphics 395 In thi
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Chapter Five Similarity We have sho
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Section I. Complex Vector Spaces 39
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Section I. Complex Vector Spaces 40
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Section II. Similarity 403 represen
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Section II. Similarity 405 A common
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Section II. Similarity 407 ̌ 1.13
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Section II. Similarity 409 2.4 Lemm
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Section II. Similarity 411 Exercise
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Section II. Similarity 413 where x
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Section II. Similarity 415 We next
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Section II. Similarity 417 3.12 Def
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Section II. Similarity 419 3.18 The
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Page 475 and 476: Topic: Method of Powers 465 ⃗v T
Page 477 and 478: Topic: Method of Powers 467 39368 -
Page 479 and 480: Topic: Stable Populations 469 Now i
Page 481: Topic: Page Ranking 471 importance
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
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