Are there real normal matrices with non-negative entries that are asymmetric and non-circulant?

Question

Is there an example of a normal matrix with real non-negative entries that is neither symmetric nor circulant/block-circulant?

If not, is there a proof of this property/reference to proof?

Additionally, what if the entries on the diagonal are strictly 0, does this collapse the set of non-normal matrices onto symmetric or circulant/block-circulant matrices?

Thanks

Answer 1

For example, $$ \pmatrix{0 & 0 & 1 & 0\cr 0 & 0 & 0 & 1\cr 0 & 1 & 0 & 0\cr 1 & 0 & 0 & 0\cr} $$

Or, slightly more generally, $$ \pmatrix{ b & b +r & a +s & a \cr b +r & b & a & a +s \cr a & a +s & c & c +r \cr a +s & a & c +r & c }$$