# Example: Sum of non-commuting matrices is not normal

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!

Take any matrix $A$ which is not normal. Then $A = \frac12(A+A^*) + \frac12(A-A^*)$, and $B = \frac12(A+A^*)$ and $C = \frac12(A-A^*)$ are both normal. Also they cannot commute, because $AA^*-A^*A = (B+C)(B-C) - (B-C)(B+C) = 2(CB-BC)$, and so if they did commute, then $A$ would be normal.
Construct a defective matrix, something a normal matrix never is, by $$C=\pmatrix{3& 2 \\ 0 & 3} = \pmatrix{3&1\\1&3} + \pmatrix{\phantom{-}0&\phantom{-}1\\-1&\phantom{-}0}=A+B.$$
$C$ is not normal since $[CC^\dagger,C^\dagger C]_-=\pmatrix{\phantom{-}0 & \phantom{-}48\\-48 &\phantom{-}0}$ and $A$ and $B$ don't commute as well $[A,B]_-=\pmatrix{-2 & \phantom{-}0 \\ \phantom{-}0 &\phantom{-}2}$.