If $\mathbf{Ax}=\mu\mathbf y$ and $\mathbf{Ay}=\mu\mathbf x$, are $\mu$ or $-\mu$ an eigenvalue of $\mathbf A$?

Question

Combining the two gives $\mathbf A^2\mathbf x = \mu^2\mathbf x$, but this does not necessarily imply that $\pm\mu$ are eigenvalues of $\mathbf A$ (rotation by $90^\circ$ in the plane comes to mind).

I appreciate any assistance with this.

Answer 1

Second Hint : Try taking a look at $A\mathbf x + A\mathbf y$ and $A\mathbf x - A\mathbf y$

Answer 2

Hint: Matrix equations are the natural home of linear combinations of things ...