Around the year 1202, the mathematician Fibonacci wrote down a famous sequence of numbers when describing the breeding pattern of pairs of rabbits. The Fibonacci sequence is Recursively the Fibonacci sequence is ao = 0, a₁ = 1, an+1 = an+an-1 for n ≥ 1. In this exercise, we will derive a beautiful formula for the n-th Fibonacci number using eigenvalues, eigenvectors, and matrix powers! To begin, observe that the following matrix equation is true: (a) Let's work with Equation 1 when n = 3. Notice that by the recursion formula, we have: @=[ JQ-E J(J·D=J · J (J). = (b) Let A = Following a similar argument, show that This generalizes! For all n ≥ 1, we have that 1-610 0, 1, 1, 2, 3, 5, 8, 11, 19,,.... 1- and [¹]=[][](Equation 1) an-1 (Equation 2) You can take Equation 2 for granted, though it's a direct generalization of the work you've done! [J √5 2 Av=Xv.). [an+1] an sic -eigvenctor is 5 B-JO = Explain why this is enough to conclude that A is diagonalizable. 1+ √5] = (c) Let's denote the eigenvalues in part (b) as + and o- 2 number + is called the golden ratio and is one of the most bers in mathematics! Show that a basic +-eigenvector is = Find CA (x) and verify that that the two eigenvalues of A are [1] 1+ √5] 2 1- √5 The 2 famous num- and that a ba- (you don't need to solve systems, just use the equation
Around the year 1202, the mathematician Fibonacci wrote down a famous sequence of numbers when describing the breeding pattern of pairs of rabbits. The Fibonacci sequence is Recursively the Fibonacci sequence is ao = 0, a₁ = 1, an+1 = an+an-1 for n ≥ 1. In this exercise, we will derive a beautiful formula for the n-th Fibonacci number using eigenvalues, eigenvectors, and matrix powers! To begin, observe that the following matrix equation is true: (a) Let's work with Equation 1 when n = 3. Notice that by the recursion formula, we have: @=[ JQ-E J(J·D=J · J (J). = (b) Let A = Following a similar argument, show that This generalizes! For all n ≥ 1, we have that 1-610 0, 1, 1, 2, 3, 5, 8, 11, 19,,.... 1- and [¹]=[][](Equation 1) an-1 (Equation 2) You can take Equation 2 for granted, though it's a direct generalization of the work you've done! [J √5 2 Av=Xv.). [an+1] an sic -eigvenctor is 5 B-JO = Explain why this is enough to conclude that A is diagonalizable. 1+ √5] = (c) Let's denote the eigenvalues in part (b) as + and o- 2 number + is called the golden ratio and is one of the most bers in mathematics! Show that a basic +-eigenvector is = Find CA (x) and verify that that the two eigenvalues of A are [1] 1+ √5] 2 1- √5 The 2 famous num- and that a ba- (you don't need to solve systems, just use the equation
Chapter12: Sequences, Series And Binomial Theorem
Section: Chapter Questions
Problem 327PT: Find the twenty-third term of an arithmetic sequence whose seventh term is 11 and common difference...
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