Square Root of Sample Covariance Matrix, the Component-Wise Formulation of

Question

Let $X \in \mathbf R^{n \times p}, n<p$ be a given matrix. Let us form the square matrix $M \in \mathbf R^{n \times n}$ such that $MM^T = XX^T.$ (This is done by writing down the diagonalisation $XX^T = A^T D A$, taking $\sqrt D$ as the entry-wise square root of $D$, and then defining $M:= A^T \sqrt D$.) Is it possible to describe the function $f$ such that

$$M_{i,j} = f(X_i,X_j)?$$

Thanks!