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**Opalg** If $\displaystyle A = e\otimes f + f\otimes e$ then $\displaystyle Ae = e\langle f,e\rangle + f\langle e,e\rangle$ and $\displaystyle Af = e\langle f,f\rangle + f\langle e,f\rangle.$ On the two-dimensional subspace spanned by e and f, the matrix of A (with respect to the basis {e,f}) is $\displaystyle M = \begin{bmatrix}\langle f,e\rangle&\langle f,f\rangle\\ \langle e,e\rangle&\langle e,f\rangle\end{bmatrix}.$

Find the eigenvalues and eigenvectors of that, just as you would for any 2x2 matrix, by solving the equation $\displaystyle \det(M-\lambda I) = 0.$

Note: if e and f are linearly dependent then the answer will be different (but easier).