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.

Note. In fact, we're proving that stable subspaces have a stable orthogonal.


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