Math254Eigenvalues

Math254Eigenvalues - MATH 254 NOTES ON 7.1 How do we find...

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MATH 254: NOTES ON 7.1 How do we find eigenvalues for large matrices? If a matrix is upper or lower triangular, its eigenvalues are simply the entries along the main diagonal. In Example 6 on pp.386-7, we get relatively lucky with the matrix A . Cofactor expansions can be used to expand l IA - . If you exploit “0”s along the way, the expansion is quick and easy. It turns out that - is simply the product of the diagonal entries of - . Of course, we’re not always so lucky! 7.1, #21 We will find the eigenvalues of A = - -- È Î Í Í Í ˘ ˚ ˙ ˙ ˙ 03 5 44 1 0 00 4 . This problem is similar to Example 8 on p.389. We luck out in that the third row has a couple of “0”s, so we can use it as our “magic row” in our cofactor expansion. ll lll -= - - - =+ - () - =- - - [] ()-- - + 35 44 1 0 4 4 3 1 2 1 2 462 2 characteristic polynomial 12 444 3 The eigenvalues of A are the roots of its characteristic polynomial: 4, 6, and - 2 . SEE BACK

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7.1, #19 We will find the eigenvalues of A = - -- È Î Í Í Í ˘ ˚ ˙ ˙ ˙ 12 2 25 2 66 3 .
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Math254Eigenvalues - MATH 254 NOTES ON 7.1 How do we find...

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