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Say I have some mapping in 2D t(v) = ... that is a tensor. I can find a matrix 2x2 T that represent this tensor, and find the eigenvalues and the eigenvectors of this matrix.

I've been given a task by my lecturer: Transform this matrix T to the eigen system determined by the normalized eigenvectors.

As I understand it, I should transform this matrix to get the new coordinate system, somehow represented by the eigenvectors. The transformation itself isn't difficult: QTQ'.

What I can seem to figure out is: how do I get the rotation matrix Q?

Any help would be greatly appreciated, Paul

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Can you provide a sample matrix $T$? –  Amzoti Jan 28 '13 at 18:49
    
T = [ -0.28 0.96 , 0.96 0.28 ] –  PawelMysior Jan 28 '13 at 18:56

1 Answer 1

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If you write matrix multiplication as $Tv$, according to the notation $t(v)$ for the corresponding mapping, then the columns of $Q'$ will be just the (normalized) eigenvectors, and $Q=(Q')^{-1}$.

(In case, the eigenvectors are orthogonal and normailzed, we also have $Q'=Q^T$.)

Anyway, for this specific transformation you don't even need to calculate $Q$ and $Q'$, as the result will be the diagonal matrix with the corresponding eigenvalues in the diagonal. Can you see why?

(How is the matrix of a given linear transformation defined/constructed in general with respect to a given basis?)

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Hmm, I cannot see why the result should be a diagonal matrix with the eigenvalues in the diagonal. It seems that, to achieve that, I have to define the eigenvectors (the ratio between parameters) in a specific way. It's a bit confusing I have to say. –  PawelMysior Jan 28 '13 at 21:30

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