find linear approximation of non-linear matrix transform [closed]

Question

I have a square matrix denoted as $A$ and an element-wise square operator $\sigma$, such that $\sigma(A)=a_{ij}^2$,$\forall i,j$, $a_{ij}$ is the ith row and jth column element of $A$. Is there exists linear operation to approximate this non-linear operator, like $\sigma(A) \approx PAQ$ or any other form of linear matrix operation(e.g taylor expansion)? More generally condition, is there exists any linear operation to approximate some element-wise convex transform(e.g. square, cube)? Sorry, some of my words may be inappropriate because of my poor math basic. Thank you very much.