Characterising all normal matrices in $M_2(\mathbb R)$

Question

Characterise all normal matrices in $M_2(\mathbb R)$

Motivated from this post, and José Carlos Santos really nice answer to it, I have made this post.

It is my first time making such a post so I don't really know if it would be allowed.

I have nothing to add regarding the characterisation.

It would be really nice to have a characterisation for all $2\times2$ real normal matrices as it would make coming up with counter examples to certain problems quite easy.

José Carlos Santos · Accepted Answer · 2019-11-13T13:37:42.337

2

A matrix $\left[\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right]$ is normal if and only if$$\left\{\begin{array}{l}b^2=c^2\\(b-c)(a-d)=0.\end{array}\right.$$It is not hard to see that therefore the $2\times2$ normal matrices are:

The symmetric matrices.
The matrices of the type $\left[\begin{smallmatrix}a&b\\-b&a\end{smallmatrix}\right]$.

edited Nov 13 '19 at 13:37

answered Nov 13 '19 at 13:33

José Carlos Santos

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Diagonal matrices are a subset of symmetric ones, why make them explicit? – lisyarus Nov 13 '19 at 13:36
That was silly. I've edited my answer. Thank you. – José Carlos Santos Nov 13 '19 at 13:38
Thank you, I thought I'm missing something. – lisyarus Nov 13 '19 at 13:41

Characterising all normal matrices in $M_2(\mathbb R)$

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