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Lecture 36.pdf

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MATH 136
Robert Sproule

Friday, April 4 − Lecture 36 : Diagonalizable matrices : Applications Concepts : Apply the matrix diagonalization process in various problems. 36.1 Application of matrix diagonalization − The given matrix A:was diagonalized previously. Use it’s diagonalization to compute A .5 −1 Solution: We found above the diagonalizing matrix X, X and D.  Then A = XD X . It follows that: This example is a particular case of a more general statement: 36.1.1 Proposition − Let A be diagonalizable matrix with diagonalizer P and diagonal matrix D = P AP. Then for any positive integer n, A = PD P . n n −1 36.2 An application of eigenvalues to a number theory question: Consider the special sequence of numbers 1, 1, 2, 3, 5, 8, 13, ..... where each number is the sum of it’s two predecessors x = x + x . k + 2 k + 1 k − This is called a Fibonacci sequence. − Question: Is it possible to find the 1000 and 1001 term to get the 1002 nd term without finding the previous 1000 terms of this sequence? − The tools just developed on eigenvalues and eigenvectors allow us to do this. Let x0= 1 and x =11 be the first two terms of the Fibonacci sequence (described above) { x0, x1, x3, 4 ,, }. Define the vector That is
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