I need some help analyzing the variety of normal matrices $M M^\dagger -M^\dagger M = 0$. Each entry in this equation satisfies a quadratic equation $$\sum_j h_{ij} \overline{h_{kj}} = \sum_j \overline{h_{ij}} h_{kj} $$ Where $i,j = 1, \dots, n$. This reminds me of the Plucker embedding of the Grassmanian as the intersection of quadratics.

If this were a single non-degenerate conic it would be similar to some kind of cone $x_1^2 + \dots + x_k^2 - x_{k+1}^2 -\dots -x_n^2=0 $. But this is an intersection of (possibly degenerate) quadrics. Not at all sure the structure of this variety.

Is this variety reducible or singular? What kind of components does it have? What's it's dimension? I'm looking for general information about this variety.

I need some help analyzing the variety of normal matrices $M M^\dagger -M^\dagger M = 0$. Each entry in this equation satisfies a quadratic equation $$\sum_j h_{ij} \overline{h_{kj}} = \sum_j \overline{h_{ij}} h_{kj} $$ Where $i,j = 1, \dots, n$. This reminds me of the Plucker embedding of the Grassmanian as the intersection of quadratics.

If this were a single non-degenerate conic it would be similar to some kind of cone $x_1^2 + \dots + x_k^2 - x_{k+1}^2 -\dots -x_n^2=0 $. But this is an intersection of (possibly degenerate) quadrics. Not at all sure the structure of this variety.

I'm looking for general information about this variety. Is this variety reducible or singular? What kind of components does it have? What's it's dimension?

EDIT: By the spectral theorem normal matrices are similar to diagonal matrices, $\mathrm{diag}(\lambda_1, \lambda_2, \dots, \lambda_n)$ by a unitary matrix so there is a $U(n)$ action on the variety of normal matrices fibered by the diagonal matrices themselves. Some of these orbits are degenerate (like when some $\lambda = 0$).

So maybe $\dim \{ [M, M^\dagger]=0\} = \dim U(1)^n + \dim U(n) = n^2 + n$