Orthogonal Diagonalisation (Maths - Matrices)

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JimBob
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#1
Report Thread starter 17 years ago
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Doing a F.Maths question on these at the moment:

Told to find a matrix P such that P(transpose)AP diagonalises A. (3X3 Matrices)

I know the method to finding the columns of the matrix P, that is find eigen vectors (in unit vector form, because it must be orthogonal) of A to give the colums of P, but was wondering if it matters what order the columns go in?
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Jonny W
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Report 17 years ago
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If P = (u | v | w) then P^T A P =
a 0 0
0 b 0
0 0 c
where a, b, c are the eigenvalues corresponding to the eigenvectors u, v, w respectively.

So P^T A P is diagonal whichever column-order you choose for P.
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Rich
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#3
Report 17 years ago
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(Original post by JimBob)
Doing a F.Maths question on these at the moment:

Told to find a matrix P such that P(transpose)AP diagonalises A. (3X3 Matrices)

I know the method to finding the columns of the matrix P, that is find eigen vectors (in unit vector form, because it must be orthogonal) of A to give the colums of P, but was wondering if it matters what order the columns go in?
No ordering doesn't matter. However for some applications of this theory (such as reducing a conic/quadric to 'standard form'), one might be required to ensure that your isomorphism P is a particular operation (such as a rotation, det(P) = 1, as opposed to a reflection, det(P)=-1 etc.), in which case you would have to be careful as to the ordering of the eigenvectors, but if you are just diagonalising, it's of no consequence (eigenvalues in the diagonal matrix are just in the respective order of the eigenvectors).
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JimBob
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#4
Report Thread starter 17 years ago
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Thats what I was hoping, thanks guys.
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