I am interested in having a notation similar to block-matrix operations, where the elements of the block differ from each other, making it not truly a block-matrix. Its best to give an example (from my application: dynamical systems).
Say we have a vector of generalized coordinates $\mathbf{q}$ for a floating-base robot. That means I can separate the coordinates into base configuration ---member of the special euclidean group--- $\mathbf{X}_B \in \mathcal{SE}(3)$ and joint-space $\mathbf{\hat{q}} \in \mathcal{R}^n$. I would like to define my vector of generalized coordinates in block notation $\mathbf{q} = [\mathbf{X}_B \quad \mathbf{\hat{q}}]^T$ (Note here, mentioning the abuse of notation), because it is in my intereset to have a notation that can allow me to decouple the operations affecting both spaces, that is, for example:
$$ \begin{bmatrix} a_1 \cdot \mathbf{X}_B \\ \mathbf{\hat{q}} \end{bmatrix} \doteq \begin{bmatrix} a_2 \cdot \mathbf{X}_B \\ b_2 \cdot \mathbf{\hat{q}} \end{bmatrix} $$
Now because the dimensions of $\mathbf{X}_B$ and $\mathbf{\hat{q}}$ differ, what is the best notation in your opinion to do as I did. Is there a better way?