I want to transform a permutation matrix so that the resulting matrix has the elements at the same location, but contains smaller values depending on the deviation from the identity matrix.
For the displacement of the element by n rows from the identity matrix, the element should hold the value $(\frac{1}{2})^n$ in the transformed matrix.
Here are a few examples:
$$ A= \left(\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}\right) \rightarrow A'= \left(\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}\right)$$
$$ B= \left(\begin{matrix}1&0&0\\0&0&1\\0&1&0\end{matrix}\right) \rightarrow B'= \left(\begin{matrix}1&0&0\\0&0&\frac{1}{2}\\0&\frac{1}{2}&0\end{matrix}\right)$$
$$ C= \left(\begin{matrix}0&1&0\\0&0&1\\1&0&0\end{matrix}\right) \rightarrow C'= \left(\begin{matrix}0&\frac{1}{2}&0\\0&0&\frac{1}{2}\\\frac{1}{4}&0&0\end{matrix}\right)$$
How can such a transformation be realized mathematically?
In terms of the Hadamard product, the output for a given permutation matrix $A$ is $A \odot M$ where $$ M = \pmatrix{1&1/2&1/4\\1/2&1&1/2\\1/4&1/2&1}. $$