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# ws10 - Math 464 Worksheet 10 The goal in this worksheet is...

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Math 464, Worksheet 10 The goal in this worksheet is to compute a high power of a matrix. For example, bracketleftbigg . 95 . 1 . 1 . 89 bracketrightbigg 500 The process involves eigenvalues and eigenvectors , which I will define by example as they come up, and them more formally in the next worksheet. Part I. The first stage is to solve the system of two equations bracketleftbigg . 95 - λ . 1 . 1 . 89 - λ bracketrightbiggbracketleftbigg x y bracketrightbigg = bracketleftbigg 0 0 bracketrightbigg for the three variables x , y and λ . That’s more than 2 variables, so we don’t use normal algebra to solve this. Instead: Any solution with x = 0 and y = 0 is forbidden. Other solutions exist if and only if the determinant of bracketleftbigg . 95 - λ . 1 . 1 . 89 - λ bracketrightbigg is zero. Therefore: 1. Solve the equation ( . 95 - λ )( . 89 - λ ) - ( . 1)( . 1) = 0 Each solution is called an eigenvalue . 2. Pick one of the eigenvalues, say λ 1 = 1 . 0244. Find one non-zero solution to the system of equations bracketleftbigg . 95 - 1 . 0244 . 1 . 1 . 89 - 1 . 0244 bracketrightbiggbracketleftbigg x y

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