Let $A \in \mathbb{R}^{p \times p}$ and $B \in \mathbb{R}^{q \times q}$ be nonsymmetric matrices.
If $p=q$, $A + B$ is the canonical sum.
If $p \ne q$ the following notions exist:
The direct sum of $A \oplus B$ is \begin{align} \pmatrix{A & 0 \\ 0 & B} \end{align}
The subdirect sum $A \oplus_k B$, where $k$ is the level of overlap. For instance if $p=q=4$ and $k = 2$: \begin{align} \pmatrix{a_{11} & a_{12} & a_{13} & a_{14} & 0& 0\\ a_{21} & a_{22} & a_{23} & a_{24}& 0& 0\\ a_{31} & a_{32} & a_{33} + b_{11} & a_{34} + b_{12} &b_{13}&b_{14}\\ a_{41} & a_{42} & a_{43} + b_{21} & a_{44} + b_{22} &b_{23}&b_{24}\\ 0 & 0 & b_{31} & b_{32} &b_{33}&b_{34} \\ 0 & 0 & b_{41} & b_{42} &b_{43}&b_{44}} \end{align}
Define the numerical range or field of values as \begin{align} W(A) = \left\{\frac{v^ T A v}{v^T v}, 0 \ne v \in \mathbb{C}^{p + q - 2k} \right\} \end{align} the following properties are known: \begin{align} W(A + B) \subseteq& W(A) + W(B) \\ W(A \oplus B) =& Co(W(A) \cup W(B)) \end{align} where $Co()$ is the convex hull of a set.
What can be said about $W(A \oplus_k B)$?