Lecture05-2003 - Lecture IV Eigenvalues, Eigenvectors, and...

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Lecture IV Eigenvalues, Eigenvectors, and Norms I. Eigenvalues and Eigenvectors Just to make sure that you haven’t picked up any bad habits, the determinant of any n * n matrix can be derived by expanding down any column or across any row of the matrix: aaaa a aaa a a a 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 11 11 22 23 24 32 33 34 42 43 44 21 21 12 13 14 23 33 34 42 43 44 31 31 12 13 14 22 23 24 42 43 44 41 41 12 13 14 22 23 23 32 33 34 1 1 =− +− +++ + () where a aa a a 22 23 24 32 33 34 42 43 44 22 33 34 43 44 32 23 24 42 43 42 23 24 33 34 =−+ . The eigenvalues of a matrix A are then defined as the solutions to the equation AI −= λ 0 where I is the identity matrix. One example of the use of eigenvalues is from differential equations: dv dt vw v a t t dw wa t t = = = = 45 8 0 23 8 0 . Another way to write this problem is ut vt wt uA ,, . = = = 0 8 5 22
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AEB 6533–Static and Dynamic Optimization Lecture 5 Professor Charles B. Moss 2 Thus, du dt Au t u u == (), () 0 0 . We are looking for solutions of the form ut e y wt e z t t () = = λ . Substituting this solution into the system of differential equations, we have 45 23 tt t t ey ey ez ez λλ λ λ  λ− =   . next, we divide through by e λ t yielding y z yz = or yy zz xA x  λ=   which is the definition of the eigenvalues, λ , and the eigenvectors, x , Ax x AI x = −= 0 0 where the eigenvalues of x lie in the null space of the matrix −− .
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Lecture05-2003 - Lecture IV Eigenvalues, Eigenvectors, and...

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