I'm trying to find an example of two non-commuting normal matrices, such that their sum is not normal. I know that unitary, orthogonal, hermitian and symmetric matrices are all normal.
I figure it can't be the sum of symmetric or hermitian matrices, because they would be symmetric (hermitian) again. I also know that 2x2 orthogonal matrices are also symmetric so their sum wouldn't be an example either.
Any help on which characterstics I could play with to find such an example would be greatly appreciated!