# Is there any constraint for the mixed effects design matrix $Z$?

In the mixed effects models, we have

$$y = X\beta + Zu + \epsilon$$

where $$u$$ : mixed effects unknown vector, and $$Z$$: mixed effects design matrix.

I assume $$\mathbb{u}_p \sim N(0, \sigma I_p)$$ and would like to know how much freedom I have with respect to the choice of $$Z$$.

I know that in many cases $$Z = \begin{bmatrix} 1_{q_1} & 0 & 0 &\cdots & 0 \\ 0 & 1_{q_2} & 0 &\cdots & 0 \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 &\cdots & 1_{q_p} \\ \end{bmatrix}$$ where $$q_i$$'s are vectors of $$1$$.

But is there any constraint that prevents me from choosing $$Z$$ other than the above? Any example will be appreciated

In general, you do not have any restrictions in the specification of the $$Z$$ matrix, but making it too complex often makes the model unstable.