# Normal matrix as sum of normal matrices

If

$$N_1$$ and $$N_2$$ are normal matrices, and they commute,

Then

$$N=N_1+N_2$$ is also normal.

Does the reverse implication hold? it is if a normal matrix $$N$$ can be expressed as the sum of two normal matrices $$N_1$$ and $$N_2$$, is it true that these commute ?

In general, no. The matrices$$N_1=\begin{bmatrix}1&2\\2&2\end{bmatrix}\text{ and }N_2=\begin{bmatrix}2&2\\2&1\end{bmatrix}$$are normal and don't commute, but$$N_1+N_2=\begin{bmatrix}3&4\\4&3\end{bmatrix}$$is normal too.