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    A = \[ \left( \begin{array}{ccc}

6 & 0 & 2 \\

0 & -1 & 0 \\

2 & 0 & 3 \end{array} \right)\]

    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.
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    (Original post by shawn_o1)
    A = \[ \left( \begin{array}{ccc}

6 & 0 & 2 \\

0 & -1 & 0 \\

2 & 0 & 3 \end{array} \right)\]

    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)
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    (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.
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    (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 e_1, e_2, e_3, we have A.e_1 = -e_1, A.e_2 = 2 e_2, A.e_3 = 7e_3. Does that ring any change-of-basis bells?
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    oh, now I see where I went wrong, I tried to multiply A with

    \[ \left( \begin{array}{ccc}

0 & 1 & 0 \\

1 & 0 & -2 \\

2 & 0 & 1 \end{array} \right)\]

    when that was actually the transpose. thanks
 
 
 
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