Matrices with eigenvalues in geometric progression Given $a\in\Bbb F$ are there any natural $k\times k$ matrices in $\Bbb F^{k\times k}$ with $a^{0},a^1,a^2,\dots, a^{k-1}$ as eigenvalues where $\Bbb F$ is any char $0$ field?
The characteristic equation should have roots in G.P. as well. How should the char equation look like? 
Can we conclude that all such matrices should be structurally similar and just look at circulants with these prescribed eigenvalues since other matrices are just conjugate by unitaries of diagonal matrices with specified numbers as roots? (Note circulants are rotations by FFT)
What is the matrix structure in general?
 A: If you have $k$-by-$k$ real matrices then the characteristic polynomial must be real. Assume that the eigenvalues are real and all have a common ratio. If they are all real then:


*

*$\chi(s) \propto (s-v)^k$

*$\chi(s) \propto (s-v)(s-rv)(s-r^2v)\ldots (s-r^kv)$


In the first case, the matrix is a multiple of the identity. In the second it is diagonalisable.
If there is a complex eigenvalue then its conjugate must also be an eigenvalue. If the common ratio is $r$ then there exists a positive integer $k$ for which 
$$z=r^kz^* \implies |z|=|r^kz^*| \implies |z|=|r|^k|z^*| \implies |r|^k=1 \implies |r|=1$$
It follows that the eigenvalues must all have the same modulus: $|r^kz|=|r|^k|z|=|z|$. Moreover, since they appear in pairs, there must be an even number of them. Hence, the characteristic polynomial must be of the form
$$\chi(s) \propto (s-z_1)(s-z_1^*)\ldots(s-z_m)(s-z^*_m)$$
$$\chi(s) \propto (s^2-2\Re(z_1)s+\ell^2)\ldots(s^2-2\Re(z_m)s+\ell^2)$$
where $\ell$ is the common modulus of the roots which have a common ratio.
