There are advantages to working with complex numbers.
- Proof
- Shur’s identity can be rewritten as ( unitary and triangular). Then
normal implies Recalling that , and that unitary means , the above identity
simplifies to One can check that this last condition implies must be a diagonal
matrix (in other words, the only triangular matrix which is also normal is a
diagonal one. Note, though, that need not have real entries).
Conversely, if is diagonal, then is normal (as we just noted), and is unitarily similar to a normal matrix, which implies it too is normal.