For a fixed k and positive semidefinite n x n matrix A with rank(A) = r > k, is it always possible to express A as a sum of matrices of the form XX\^T, such that rank(XX\^T) = rank(X) = k and X is n x k?

I've thought about this for a while, and the best I can manage is to write A as LL\^T where L is n x k by using the spectral decomposition and removing rows/columns from the diagonal and orthonormal matrix corresponding to zero eigenvalues/vectors,, but L is still rank r, not k