What is the solution for the matrix $X$ in the following quadratic matrix equation?
$$ XFX - F^{-1}X^{-1}F = 0 $$
where $F$ is a $N \times N$ discrete Fourier transform (DFT) matrix, so it's unitary ($F^{-1} = F^\dagger$). Based on similar quadratic matrix equations it seems that $X$ would be expressed in terms of fractional powers of $F$. I have access to the eigenvectors of $F$ and hence all fractional powers of $F$ thanks to the discrete fractional Fourier transform (DFRT) in Candan et al. (2000).
Found an almost trivial solution $X = F^{-1/3}$ by guessing and checking.
Substituting $X = F^{-1/3}$ into the equation results in $F^{1/3} - F^{1/3} = 0$ which is true hence $X = F^{-1/3}$ is a valid solution.
Are there other nontrivial solutions? $F^{-1/3 + 4n}$ for integer $n$ is another obvious one due to the periodicity of the fractional Fourier transform.
The equation is equivalent to $$ (XF)^3=XF(XFX)F=XF(F^{-1}X^{-1}F)F=F^2. $$ Therefore the solutions are given by $X=CF^{-1}$ where $C^3=F^2$.
Remark. Since the eigenvalues of $F$ are $\pm1$ and $\pm i$, the matrix $C$ (which is a cubic root of $F^2$) in general doesn't commute with $F$ (which is a square root of $F^2$). Therefore $CF^{-1}$ is not necessarily an inverse cubic root of $F$.