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orthogonal matrix question

Unparseable latex formula:

A = \[ \left( \begin{array}{ccc}[br]6 & 0 & 2 \\[br]0 & -1 & 0 \\[br]2 & 0 & 3 \end{array} \right)\]



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.
Reply 1
Avatar for Smaug123
Smaug123
15
Original post by shawn_o1
Unparseable latex formula:

A = \[ \left( \begin{array}{ccc}[br]6 & 0 & 2 \\[br]0 & -1 & 0 \\[br]2 & 0 & 3 \end{array} \right)\]



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)
Reply 2
Avatar for shawn_o1
shawn_o1
OP
21
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.
Reply 3
Avatar for Smaug123
Smaug123
15
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 e1,e2,e3e_1, e_2, e_3, we have A.e1=−e1,A.e2=2e2,A.e3=7e3A.e_1 = -e_1, A.e_2 = 2 e_2, A.e_3 = 7e_3. Does that ring any change-of-basis bells?
Reply 4
Avatar for shawn_o1
shawn_o1
OP
21
oh, now I see where I went wrong, I tried to multiply A with

Unparseable latex formula:

\[ \left( \begin{array}{ccc}[br]0 & 1 & 0 \\[br]1 & 0 & -2 \\[br]2 & 0 & 1 \end{array} \right)\]



when that was actually the transpose. thanks

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