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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 ?

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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.

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  • $\begingroup$ +1: How did you come up with that example? $\endgroup$ – copper.hat Feb 19 at 21:20
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    $\begingroup$ I picked two symmetric matrices. Automatically, both of them are normal and so is their sum. And since matrices seldom commute… $\endgroup$ – José Carlos Santos Feb 19 at 21:23
  • $\begingroup$ Yes, thanks, it was that simple. And it's always never granted to see the simple in any system or question. $\endgroup$ – orazioster Feb 22 at 18:07
  • $\begingroup$ If my answer was useful, perhaps that you could mark it as the accepted one. $\endgroup$ – José Carlos Santos Feb 22 at 18:08

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