In $\mathbb{R}^{n}$, assume $P_{1}$ and $P_{2}$ are both positive definite. Let $A = P_{1}(P_{1} + P_{2})^{-1}$.
Can we conclude the following,
Given every vector $x$, we have \begin{equation*} |(Ax)_{i}| \leq |x_{i}|, \quad i = 1,\ldots,n \end{equation*} where $(Ax)_{i}$ denote the $i$th component of $Ax$.
Is the statement above true? If not, what can we add to make it true.
P.S., This problem may be solved if we can check the infinity norm of $A$ is less than 1, but I can only say all the eigenvalues of $A$ are between $0$ and $1$.
Any suggestions? Thank you in advance.