# Normal Block Upper Triangular Matrix

What is an economical proof that a block upper triangular matrix is normal iff its off-diagonal blocks is zero and each of its diagonal blocks is normal?

By economical proof I mean a short proof, one that takes a few lines at most to write down.

Write $$M = \begin{pmatrix} A & B \\ 0 & C \end{pmatrix}$$ $$M^* = \begin{pmatrix} A^* & 0 \\ B^* & C^* \end{pmatrix}$$ If $M$ is normal, doing the block-computations gives you $AA^* + BB^* = AA^*$, so $BB^* = 0$, so $B=0$. Now the fact that $A$ and $B$ are normal is obvious.
Apply this result to $B$ for an immediate recurrence on the dimension.