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# How are eigenvalues helpful in finding invariant lines? watch

1. I had a question, that asked me to find the eigenvalues of a matrix and HENCE find the invariant lines of that matrix.

Well I used the usual method of Det[A - Lambda.I] = 0 to find the eigenvalues.

I then used the fact that invariant lines pass through the origin => c = 0, thus they're of the form y = mx, so on these lines a general point (t, mt) is transformed to the point (T, mT). So we get: Matrix . (t, mT) = (T, mT) and got simulatenous equations and solved for two values of m.

But, how do I use eigenvalues to find these invariant lines?

Thanks
2. Invariant lines correspond to eigenvectors. If you find an eigenvector (x, y) then the line through the origin and (x, y) will be invariant.
3. Right, thanks, that was pretty damn obvious since eigenvectors by definition lie on invariant lines. So I find the gradients of the two eigenvectors from my two eigenvalues, and hence y_1 = m_1.x and y_2 = m_2.x

Cheers.

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Updated: April 11, 2006
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