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Find an orthogonal basis for $\mathbb R^3$ consisting of the eigenvectors of the matrix $$\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$$

Isn't this question basically just asking 'find the eigenvectors of this matrix'? And the part about finding 'an orthogonal basis' is irrelevant?

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    The insistence on orthgonality is *not* irrelevant. If some eigenspace has dimension greater than 1 then you need to be careful to pick a basis for that eigenspace consisting of mutually perpendicular eigenvectors. It definitely isn't always going to work if you don't make a good choice. To take an extreme case, try the 3x3 identity matrix: any nonzero vector is an eigenvector but it's far from true that all choices of eigenbases are mutually perpendicular. In your matrix, one eigenvalue has multiplicity 2, so the requirement that your basis be orthogonal has content.2012-04-22
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    The question tells me that this matrix contains a basis for $\mathbb R^3$ so I know it has 3 eigenvectors. It's not like I then have a choice of eigenvectors...I just work out the eigenvectors - as (1,1,1), (-1,1,0) and (-1,0,1) - and that's it. So I can't see what relevance there is to specifying orthogonality in this question?2012-04-22
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    @KCd OK I see you are correct. I dotted those vectors are and they are not orthogonal. So how can I get an orthogonal basis?2012-04-22

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