Page "Normal matrix" Paragraph 21

The spectral theorem for normal matrices can be seen as a special case of the more general result which holds for all square matrices: Schur decomposition.

In fact, let A be a square matrix.

Then by Schur decomposition it is unitary similar to an upper-triangular matrix, say, B.

If A is normal, so is B.

But then B must be diagonal, for, as noted above, a normal upper-triangular matrix is diagonal.