Yahoo Answers: Answers and Comments for Linear algebra!! HELP? [Mathematics]
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From Anonymous
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Thu, 20 Jun 2013 21:45:38 +0000
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Yahoo Answers: Answers and Comments for Linear algebra!! HELP? [Mathematics]
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https://ca.answers.yahoo.com/question/index?qid=20130620214538AAtjyRE
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From kb: First, find the eigenvalues of A by solving A...
https://ca.answers.yahoo.com/question/index?qid=20130620214538AAtjyRE
https://ca.answers.yahoo.com/question/index?qid=20130620214538AAtjyRE
Fri, 21 Jun 2013 05:05:52 +0000
First, find the eigenvalues of A by solving A  λI = 0.
4λ 0 2
6 1λ 6 = 0
4 0 2λ
Expanding down the second column:
(1  λ) [(λ^2  2λ  8) + 8] = 0
==> λ = 1, 0, 2.

Next, we find corresponding eigenvectors.
For λ = 1, we solve (A  (1)I)v = 0.
[5 0 20]
[6 0 60]
[4 0 10], which reduces to
[1 0 00]
[0 0 10]
[0 0 00], yielding eigenvector (0, 1, 0)^T.

For λ = 0, we solve (A  0I)v = 0.
[4 0 20]
[6 1 60]
[4 0 20], which reduces to
[1 0 1/20]
[0 1 30]
[0 0 00], yielding eigenvector (1, 6, 2)^T.

For λ = 2, we solve (A  2I)v = 0.
[2 0 20]
[6 3 60]
[4 0 40], which reduces to
[1 0 10]
[0 1 00]
[0 0 00], yielding eigenvector (1, 0, 1)^T.

So, let D be the diagonal matrix of eigenvalues
[1 0 0]
[0 0 0]
[0 0 2],
and let P be the matrix whose columns are the corresponding eigenvectors
[0 1 1]
[1 6 0]
[0 2 1].
Then, P^(1) AP = D, as required.
I hope this helps!