TITLE: Is the sum of two normal operators normal?
QUESTION [5 upvotes]: A normal operator is defined as
   $AA^{*}\ =A^{*}A$
Where A is an operator
how do i show the sum of two normal operators is normal?  Or find a counter example that shows this is false?

REPLY [8 votes]: Try $U_1 = \begin{bmatrix} 0 & 1 \\ -1 & 0\end{bmatrix}$, $U_2 = \begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}$. Both are unitary, hence normal. However $U_1+U_2 = \begin{bmatrix} 0 & 2 \\ 0 & 0\end{bmatrix}$, and $(U_1+U_2) e_1 = 0$, but $(U_1+U_2)^* e_1 = 2 e_2$, hence $U_1+U_2$ is not normal.