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I was trying to solve a coupled torsion pendulum (note this is for an assignment, so please just guidance, no solutions), consisting of two rods suspended on a vertical wire clamped at either end. I've obtained the matrix equation $$ \begin{bmatrix} -\Omega & \Omega\\ \Omega & -\frac{1}{3}\Omega\\ \end{bmatrix} \begin{bmatrix} A\\ B\\ \end{bmatrix} = \lambda \begin{bmatrix} A\\ B\\ \end{bmatrix} $$

From here I formed the characteristic equation $\lambda^2 + \frac{4}{3}\Omega\lambda - \frac{2}{3}\Omega^2 = 0$, and found the eigenvalues $\lambda = -\frac{1}{3}\Omega(2 + \sqrt{10})$ and $\frac{1}{3}\Omega(\sqrt{10} - 2)$ (which I checked with WolframAlpha).

Then, something strange happened... when I tried to use these eigenvalues to compute the eigenvectors (both by hand and with WolframAlpha), I found I couldn't. The resulting matrices had non-zero determinants (interestingly when I first put the matrix into WolframAlpha, it gave me the eigenvalues and eigenvectors, but when I plugged those eigenvalues back in, it did not give me eigenvectors).

My question is, have I done something fundamentally wrong, or made a simple mistake? I have been over it myself but I can't find any simple errors, which leads me to believe that perhaps my approach is wrong. Honestly I'm just stumped that the eigenvalues aren't giving me eigenvectors, it doesn't seem possible.

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1 Answer 1

I suggest that you put $\Omega:=1$ for a moment and then repeat your calculations, being very careful with the signs. Then you will see that everything is in order. To find the two eigenspaces you have to determine the nullspaces of the two matrices $$\left[\matrix{-1-\lambda_1 & 1 \cr 1& -{1\over3}-\lambda_1 \cr}\right]\ ,\qquad \left[\matrix{-1-\lambda_2 & 1 \cr 1& -{1\over3}-\lambda_2 \cr}\right]$$ separately in turn.

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