A normal matrix is one such that AA*=A*A where A* denotes the conjugate transpose.
When you diagonalize a normal matrix A, the diagonal matrix contains the eigenvalues for the map.
Since these matrices are restricted to having only one eigenvalue then:
All normal nxn matrices over the Complex space are similar to a diagonal matrix that has the single eigenvalue for every entry in the diagonal.
Is this correct? Is there more that can be said? Captain. THANK YOU