Eigenvectors of sum of symmetric block matrices

Question

Suppose that we have a symmetric block matrix of the following form: $$ S = \begin{pmatrix} S_{11} & \ldots & S_{1m} \\\\ \vdots & \ldots & \vdots \\\\ S_{m1} & \ldots & S_{mm} \end{pmatrix} \in \mathbb{R}^{pnm\times pnm},$$ where $$S_{ji}=S_{ij}=\begin{pmatrix} K_{ij} & 0_{n \times n(p-1)} \\\\ 0_{n(p-1)\times n} & 0_{n(p-1) \times n(p-1)} \end{pmatrix} \in \mathbb{R}^{pn\times pn}, \quad K_{ji}=K_{ij} \in \mathbb{R}^{n\times n}$$ for $i,j=1,\ldots, m$. Furthermore, there is a second symmetric (left upper) blockmatrix $$ T = \begin{pmatrix} T_{11} & \ldots & \ldots & T_{1m} \\\\ \vdots & & ⋰ & 0 \\\\ \vdots & ⋰ & & \vdots \\\\ T_{m1} & 0 & \ldots & 0 \end{pmatrix} \in \mathbb{R}^{pnm\times pnm}$$ where $$T_{ji}=T_{ij}=\begin{pmatrix} 0_{n \times n} & B_{ij}^{12} \\\\ B_{ij}^{21} & 0_{n(p-1) \times n(p-1)} \end{pmatrix} \in \mathbb{R}^{pn\times pn}, \quad B_{ij}^{12} \in \mathbb{R}^{n\times n(p-1)}, B_{ij}^{21} \in \mathbb{R}^{n(p-1)\times n}.$$ Suppose now, that I know all eigenvalues and eigenvectors of $S$ and $T$. Can I determine the eigenvectors of the sum $S+T$ depending on the eigenvectors of $S,T$, respectively? Please note that when adding these two block matrices, it will never be the case that two non-zero matrices are added. I would be very thankful for hints.