There is a Hermitian matrix that can be decomposed into eigenvalues, and I hope to express the decomposed eigenvalue matrix in the form of a formula.

Question

Exist the matrix $\quad \mathbf{H}=\sum_{i=1}^P h_i \mathbf{\triangle}_{f_i} \boldsymbol{\Pi}^{l_i}$, where $\mathbf{\Pi}=\left[\begin{array}{cccc}0 & \ldots & 0 & 1 \\ 1 & \ldots & 0 & 0 \\ \vdots & \ddots & \ddots & \vdots \\ 0 & \cdots & 1 & 0\end{array}\right]_{\mathrm{N} \times \mathbf{N}}$, $\triangle_{f_i}=\operatorname{diag}\left(e^{-i 2 \pi f_i{ }n}, \quad n=0,1, \ldots N-1\right)$ and $h_i$ is a complex-number. $0\leq f_i\leq1，l_i\geq0$ and $l_i\in \mathbb{Z}$.
Assume that $\mathbf{A}=\mathbf{H}^H\mathbf{H}$. It is easy to know that matrix $\mathbf{A}$ is a Hermitian matrix and can be decomposed into $\mathbf{A}=\mathbf{V}\mathbf{D}\mathbf{V}^H$, where $\mathbf{V}$ is the unitary matrix and $\mathbf{D}$ is the real diagonal matrix. I hope to further express $\mathbf{D}$ and $\mathbf{V}$ in the form of matrix multiplication with respect to being the variates $h_i,f_i,l_i$.