Is there a relation between $P^*|D|P$ and $|P^*DP|$? Considering in the complex fields. Let $P$ be a nonsingular matrix, $P^* $ be its conjugate transpose, is there a relation between $P^*|D|P$ and $|P^*DP|$, where $D$ is a diagonal matrix?  In particular, is it true 
$P^* |D|P \ge |P^*DP|$ in the sense of Lowner order, or is there an order for eigenvalues?
Here $|A|=(A^*A)^{1/2}$, the absolute value of a complex matrix.
Edit As I know from Suvrit's answer, there is no relation like $P^* |D|P \ge |P^* DP|$ in the sense of Lowner order. So my question becomes, is the $i$th largest eigenvalue of 
$P^* |D|P$ larger than that of $|P^*DP|$?
 A: 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.
A: Using the same example above, I found
p=[1,1;1,2]
p =
 1     1
 1     2



d=[-1,0;0,2]


d =
-1     0
 0     2



c=[1,0;0,2]


c =
 1     0
 0     2



eig(((p*d*p)^2)^(1/2))


ans =
0.2426
8.2426



eig(p*c*p)


ans =
0.1690

11.8310
But this example shows it is possible the majorization relation holds.
