MATH 263 prep sheet

For a to be diagonalizable there must be n of them

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Unformatted text preview: ,..., v n ) for V, let w ww.prep101.com ... wn = v n − • v n , w1 v ,w v ,w w1 − n 2 w 2 − ... − n n −1 w n −1 w1, w1 w 2, w 2 w n −1, w n −1 B' is an orthogonal basis for V. The vectors B'' = (u1, u2 ,..., un ) , u = w n n , are an orthonormal wn basis for V. Diagonalizing a Matrix 1. 2. 3. Find the eigenvalues λ1,K, λn of A. Find n linearly independent eigenvectors x1, x 2 ,K, x n of A by solving ( λi I − A) x i = 0 for each eigenvalue λ . For A to be diagonalizable there must be n of them. The matrix D = P −1 AP is diagonal, where ⎡λ1 0 0 ⎤ and P = [x1 x 2 L x n ] ⎥ ⎢ 0 D=⎢ ⎢0 ⎢ ⎣0 λ2 0 0 O 0⎥ 0⎥ ⎥ λn ⎦ Solving a System of Linear ODEs by Diagonalization Given a linear system of ODEs: y1'( x ) = a11 y1 ( x ) + L + a1n y n ( x ) y 2'( x ) = a21 y1 ( x ) + L + a2 n y n ( x ) M y n '( x ) = an1 y1 ( x ) + L + ann y n ( x ) Let A = [aij ]. If A is diagonalizable with eigen...
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This document was uploaded on 03/10/2014.

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