Fix $n$ a natural number. Consider the set of all $2n \times 2n$ matrices with entries from {0,1}. This is clearly a finite set. I would like to count the number of such normal matrices for fixed $n$, call this number $N(n)$. What are the asymptotics for $N(n)$? It is easy to see that if the matrix is symmetric, then it is normal so we can get a lower bound on $N(n)$. Does anyone have any ideas on an upper bound besides the trivial one.

Since the dimension of the variety of normal matrices is the same as that of the variety of symmetric matrices (for $n\times n$ complex normal matrices, the *real* dimension of the variety is $n^2+n,$ see this discussion: structure of the variety of normal matrices) I expect that the lower bound whereof you speak is asymptotically sharp.

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