More precisely, let $u \in C^{2}(\mathbb{R}^n,\mathbb{R})$ be harmonic, $Q \in \mathbb{R}^{n \times n}$ matrix.
Classify $Q$ s.t. $u \circ Q$ harmonic.
I had a few ideas to find positive or counter examples but I have problems proving what a full classification would look like. First, orthogonal $Q$ should definitely work because the mean value property is conserved under rotation. Unless I messed up in my calculations I indeed managed to prove $\Delta (u \circ Q)=\Delta u \circ Q$ for orthogonal $Q$. Trivially, $Q=id$ and $Q=0$ satisfy the desired property as well.
However, I have trouble finding more classes of matrices that satisfy this. The closest I can get to an iff-type statement is the expression for $\Delta (u \circ Q)$ I derived when looking at orthogonal matrices:
$\Delta (u \circ Q) = \sum_{j,k=1}^{n}(\sum_{m=1}^{n}q_{ki}q_{ji})(\partial_{k}\partial_{j}u)\circ Q$
For orthogonal $Q$ and harmonic $u$ this is clearly zero. But unless I am missing something there could be a plethora of different ways this is zero for harmonic $u$.
Does anyone have tips?