How are eigenvalues helpful in finding invariant lines?

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

Reply 1

mikesgt2

Invariant lines correspond to eigenvectors. If you find an eigenvector (x, y) then the line through the origin and (x, y) will be invariant.

Reply 2

username9816

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.