Let U be a $\Bbb{R}^{(n+1)(n+1)} $ matrix with entries drawn from a independent normal distribution, e.g.
$$ U_{i j} \sim N(0,1) \quad \quad i,j=1,...n+1$$ Let $ G=U U^* $ be a Gram matrix where $ U^* $ is the conjugate transpose of U and $ (G_{yx})_{ij} = G_{i+1,j} \ $ as well as $ (G_{xx})_{i,j}=G_{i,j} $ for $ \ i,j=1,...,n .$ Finally let's construct the matrix
$$ K= G_{xx}^{(-)} G_{yx} $$
where $G_{xx}^{(-)}$ is the Penrose (pseudo) inverse of $G_{xx}.$ Note that K can be interpreted as a random Koopman operator.
The spectrum of $K$ is rather interesting: For $n=260$ for example it typically looks as follows:
where the x-axis(y-axis) describes the real(imaginary) part of the eigenvalues of K. Hence it seems like most eigenvalues lie on the complex unit circle.
Can anyone offer an explanation for this observation or can lead me in the direction of further study?
