Lesson 16 The Spectral Theorem and Applications

Lesson 16 (S&H, Section 14.6)The Spectral Theorem and ApplicationsMath 20October 26, 2007AnnouncementsWelcome parents!Problem Set 6 on the website. Due October 31.OH: Mondays 1–2, Tuesdays 3–4, Wednesdays 1–3 (SC 323)Prob. Sess.: Sundays 6–7 (SC B-10), Tuesdays 1–2 (SC 116)OutlineHatsumonConcept ReviewEigenbusinessDiagonalizationThe Spectral TheoremThe split caseThe symmetric caseIterationsApplicationsBack to FibonacciMarkov chainsA famous math problem“A certain man had one pairof rabbits together in acertain enclosed place, andone wishes to know howmany are created from thepair in one year when it is thenature of them in a singlemonth to bear another pair,and in the second monththose born to bear also.Because the abovewritten pairLeonardo of Pisain the first month bore, you(1170s or 1180s–1250)will double it; there will bea/k/a Fibonaccitwo pairs in one month.”f (1) = 1f (2) = 2f (3) = 3f (4) = 5f (5) = 8Diagram of rabbitsf (0) = 1f (2) = 2f (3) = 3f (4) = 5f (5) = 8Diagram of rabbitsf (0) = 1f (1) = 1f (3) = 3f (4) = 5f (5) = 8Diagram of rabbitsf (0) = 1f (1) = 1f (2) = 2f (4) = 5f (5) = 8Diagram of rabbitsf (0) = 1f (1) = 1f (2) = 2f (3) = 3f (5) = 8Diagram of rabbitsf (0) = 1f (1) = 1f (2) = 2f (3) = 3f (4) = 5Diagram of rabbitsf (0) = 1f (1) = 1f (2) = 2f (3) = 3f (4) = 5f (5) = 8Sof (k) = f (k − 1) + f (k − 2)An equation for the rabbitsLet f (k) be the number of pairs of rabbits in month k. Each newmonth we haveThe same rabbits as last monthEvery pair of rabbits at least one month old producing a newpair of rabbits