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# orthogonal matrix question Watch

1. I've found the eigenvalues for this matrix (-1, 2 and 7). Now I have to find an orthogonal matrix P and a diagonal matrix D such that PTAP = D. I did find the bases for the eigenspaces and yet I'm still stuck.
2. (Original post by shawn_o1)

I've found the eigenvalues for this matrix (-1, 2 and 7). Now I have to find an orthogonal matrix P and a diagonal matrix D such that PTAP = D. I did find the bases for the eigenspaces and yet I'm still stuck.
What are the eigenvectors corresponding to each eigenvalue? (that is, the span of the eigenspaces for each eigenvalue)
3. (Original post by Smaug123)
What are the eigenvectors corresponding to each eigenvalue? (that is, the span of the eigenspaces for each eigenvalue)
I worked them out as (0, 1, 0)T, (1, 0, -2)T and (2, 0, 1)T though I'm not sure I got the first one right.
4. (Original post by shawn_o1)
I worked them out as (0, 1, 0)T, (1, 0, -2)T and (2, 0, 1)T though I'm not sure I got the first one right.
Your first one is certainly right - it's easy to calculate A.(0,1,0) = (0, -1, 0).

OK, if we call those eigenvectors , we have . Does that ring any change-of-basis bells?
5. oh, now I see where I went wrong, I tried to multiply A with

when that was actually the transpose. thanks

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