MATH 263 prep sheet

# Find characteristic polynomial p det i a our course

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Unformatted text preview: s To solve f ( x ) dx 1 s 1 cos( ax ) dn y d n −1 y dy an ( x ) n + an −1 ( x ) n −1 + L + a1 ( x ) + a0 ( x ) = F ( x ) : dx dx dx − sx λi m2 mk mi is the λi : solutions x satisfying the • Eigenspace corresponding to system (λ i I − A)x = 0 • Eigenspace of a matrix: combined space of all eigenspaces corresponding to all its eigenvalues Computing Eigenvalues, Eigenvectors, and Eigenspaces 1. Find characteristic polynomial p(λ ) = det (λI − A). Our Course Booklets - free at prep sessions - are the “Perfect Study Guides.” Need help for exams? Check out our classroom prep sessions - customized to your exact course - at www.prep101.com 2. 3. 4. The eigenvalues are the roots λ1 ,L , λk of characteristic equation det (λI − A) = 0 . For each eigenvalue, solve (λi I − A)x = 0 - find a basis for the set of solutions. The set of all the basis vectors in step 3 form a basis for the eigenspace of A. Gram-Schmidt algorithm: Given a basis B' = (w1, w 2 , w 3 ,..., w n ) be the set of vectors w1 = v1 w2 = v 2 − v 2, w1 w1 w1, w1 w3 = v 3 − v 3, w1 v ,w w1 − 3 2 w 2 w1, w1 w 2, w 2 Diagonalization • Diagonal matrix A: all the elements not on the main diagonal are zero; aij = 0 if i ≠ j • Similar matrices A and B: there exists an invertible matrix P such that B = P − 1 AP o similar matrices have the same determinant, rank, nullity, and eigenvalues Diagonalizable matrix A: there exists an invertible matrix P such that D = P −1 AP is diagonal o an n x n matrix is diagonalizable if and only if the eigenspace of this matrix has n basis vectors • B = (v1, v 2 , v 3...
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