Let A and $U$ are two square matrices where A is invertible. If $$AU=UA$$ In partitioned form $$\begin{bmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{bmatrix}\begin{bmatrix}U_{11}&0_{12}\\0_{21}&U_{22}\end{bmatrix}=\begin{bmatrix}U_{11}&0_{12}\\0_{21}&U_{22}\end{bmatrix}\begin{bmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{bmatrix}$$ $\implies$ $$A_{21}U_{11}=U_{22}A_{21}$$ where $U_{11}$ and $U_{22}$ are upper triangular square matrices (dimension can be different) and $U_{11}$ and $U_{22}$ cannot have same diagonal elements
Then how to prove $A_{21}$ is a zero matrix and similarly $A_{12}$
