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Update: In the edited question, the OP asks whether $\lambda_i^\downarrow(P^*|D|P) \ge \lambda_i^\downarrow(|P^*DP|)$ (or even the reverse direction). Such a relations do not hold either. Take for e.g., $P=\begin{bmatrix} 2 & 1 \\ 2 & 2\end{bmatrix}$, and use the same $D$ as below.

Then, we have $\lambda^\downarrow(P^*|D|P) = (41.22, 0.776)$, while $\lambda^\downarrow(|P^*DP|) = (23.369, 1.369)$.

However, if one assume that $P$ is a contraction, then several interesting results can be shown.

I don't think there is any useful relation.

Here is a counterexample:

\begin{equation*} P = \begin{bmatrix} 2 & 2\\ 2 & 4 \end{bmatrix}, \end{equation*} and \begin{equation*} D = \begin{bmatrix} -2 & 0\\ 0 & 4 \end{bmatrix} \end{equation*} Then, \begin{equation*} P^T|D|P = \begin{bmatrix} 12 & 16\\ 16 & 24 \end{bmatrix} \end{equation*} and

\begin{equation*} |P^TDP|^2 = \begin{bmatrix} 640 & 1536\\ 1536 & 3712 \end{bmatrix} \end{equation*} Then, \begin{equation*} \lambda(P^T|D|P - |P^TDP|) = (-3.3678, 31.4856), \end{equation*} which is indefinite.

2 fixed bug thanx to sunni

I don't think there is any useful relation.

Here is a counterexample:

\begin{equation*} P = \begin{bmatrix} 2 & 2\\ 2 & 4 \end{bmatrix}, \end{equation*} and \begin{equation*} D = \begin{bmatrix} -2 & 0\\ 0 & 4 \end{bmatrix} \end{equation*} Then, \begin{equation*} P^T|D|P = \begin{bmatrix} 12 & 16\\ 16 & 24 \end{bmatrix} \end{equation*} and

\begin{equation*} |P^TDP| P^TDP|^2 = \begin{bmatrix} 4 640 & 0\1536\\ 0 1536 & 8 3712 \end{bmatrix} \end{equation*} Then, \begin{equation*} P^T|D|P \lambda(P^T|D|P - |P^TDP| P^TDP|) = \begin{bmatrix} 8 & 16\\ 16 & 16 \end{bmatrix} (-3.3678, 31.4856), \end{equation*} which is indefinite.

1

I don't think there is any useful relation.

Here is a counterexample:

\begin{equation*} P = \begin{bmatrix} 2 & 2\\ 2 & 4 \end{bmatrix}, \end{equation*} and \begin{equation*} D = \begin{bmatrix} -2 & 0\\ 0 & 4 \end{bmatrix} \end{equation*} Then, \begin{equation*} P^T|D|P = \begin{bmatrix} 12 & 16\\ 16 & 24 \end{bmatrix} \end{equation*} and

\begin{equation*} |P^TDP| = \begin{bmatrix} 4 & 0\\ 0 & 8 \end{bmatrix} \end{equation*} Then, \begin{equation*} P^T|D|P - |P^TDP| = \begin{bmatrix} 8 & 16\\ 16 & 16 \end{bmatrix} \end{equation*} which is indefinite.