MAS 801 It's a Discreetly Discrete World Ã¢Â€Â“ Mathematics in ... - Spms

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Eigenvalues & Eigenvectors • How can we find the eigenvalues and eigenvectors of a given n x n matrix A • We want to find λ and x such that Ax = λx • This can be rewritten as Ax - λx = 0 or (A - λI)x = 0 • Turns out that λ is an eigenvalue of A if and only if det(A - λI) = 0 The above equation is called the characteristic equation of A Example • If A = 3 1 , then det(A - λI) = 3-λ 1 = (3 - λ) 2 - 1 = λ 2 - 6λ + 8 1 3 1 3-λ • Solving the quadratic equation λ 2 - 6λ + 8 = 0 gives λ = 2 and λ = 4 • The eigenvalues of A are therefore 2 and 4
Exercise 5 • Find the eigenvalues of the 3 x 3 matrix B = 0 -1 -1 1 2 1 1 1 2
Page 1 and 2: MAS 801 It’s a Discreetly Discret
Page 3 and 4: Extreme Points • The following is
Page 5 and 6: Proving Optimality: Method of Tight
Page 11 and 12: Tight Inequalities • An inequalit
Page 13 and 14: The Web Graph • Primary focus is
Page 15 and 16: How Big is the Web (Graph)
Page 17 and 18: Searching the Web • We have seen
Page 19 and 20: Crawler • Program that browses WW
Page 21 and 22: Query Engine • When given input k
Page 23 and 24: PageRank • Assigns a rank to ever
Page 25 and 26: PageRank: First Try • We want to
Page 27 and 28: Adjacency Matrix of a Digraph • E
Page 29 and 30: Matrix Transpose Matrix Transpose
Page 31 and 32: Exercise 2 • Compute the importan
Page 33 and 34: PageRank: An Improvement • The pr
Page 35 and 36: Markov Matrix of a Digraph • Exam
Page 37 and 38: Exercise 3: Answer 1 2 3 4 5 6 •
Page 39 and 40: Exercise 4 • Compute the importan
Page 41 and 42: PageRank: An Improvement • The pr
Page 43: Some Linear Algebra • Given an n
Page 47 and 48: Example • Find all the eigenvecto
Page 49: Further Properties of Matrices •

MAS 801 It's a Discreetly Discrete World Ã¢Â€Â“ Mathematics in ... - Spms

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