M: pxp symmetric p.d. matrix with unit diagonals n: number much smaller than p

Want a nonrandom nxp matrix X such that X'X is close to M element-wise. If n gets larger, hopefully difference gets smaller.

I have a no so good method. Get the eigendecomposition M = D'VD, and rearrange the diagonals of V in decreasing order and so the rows of D. Take only the first n rows of D as X. This method works only if the eigenvalues decrease very fast.

Any input is appreciated. Thanks a lot!