$A=\begin{pmatrix}A_{11}&\dots&A_{1m}\\A_{21}&\dots&A_{2m}\\ A_{31}&\dots&A_{3m}\\\dots&\dots&\dots\\ A_{m1}&\dots&A_{mm}\end{pmatrix}$
given that each $A_{ij}$ is $n\times n$ and have a common set of indipendent eigen vector. we need to show eigenvalues of $A$ are the eigenvalues of the following matrix $\Lambda=\begin{pmatrix}\lambda^k_{11}&\dots&\lambda^k_{1m}\\\lambda^k_{21}&\dots&\lambda^k_{2m}\\ \dots&\dots&\dots\\ \lambda^k_{m1}&\dots&\lambda^k_{mm}\end{pmatrix};k=1,2,\dots,n$, $\lambda^k_{ij}$ is the $k^{th}$ eigvalues of $A_{ij}$ corresponding to the kth eigenvector $v_k$ common to all $A_{ij}$
thanks for helping