Hankel Singular Values for diagonal state-space model

Question

Let (A,B,C) be a diagonal and stable discrete-time LTI state space model: \begin{align*}x(k+1)&=Ax(k)+Bu(k),\\ y&= Cx(k)\end{align*} with A being a diagonal matrix: \begin{bmatrix} \lambda_1 &0 &\cdots&0 \\ 0 & \lambda_2&\cdots&0\\ \vdots& & \ddots &\vdots\\ 0& 0&\ldots &\lambda_n \end{bmatrix} Can we compute singular values of the Hankel matrix $\sigma_{i}(H)$ for $i=1,\ldots,n$ of this system in a closed-form, in terms of model matrices A,B,C, exploiting the diagonal A structure? Recalling that the Hankel matrix is of the form $H= \begin{bmatrix} CB &CAB &CA^2B&\cdots \\ CAB&CA^2B & CA^3B&\cdots \\ CA^2B& CA^3B & CA^4B&\cdots \\ \vdots& \vdots&\vdots & \end{bmatrix} $