How to find Basis vectors of a matrix $X$, given basis vectors of its kernel matrix $XX^T$?

Question

If we know basis vectors for $K=XX^T$ (e.g. will be eigenvectors here since $K$ is symmetric), how can we find base vectors for $X$?

Answer 1

Not generally, for example, if $X$ is orthogonal, then $X X^T = I$, and we can choose any basis we want for $I$, but that tells us nothing about $X$.

I misunderstood the question:

Since we have ${\cal R}(X) = {\cal R}(X X^T) = {\cal R}(K) $ (since ${\cal R}(X^T) = \ker X^\bot$), any basis for ${\cal R}(K)$ will do, for example, any maximal linearly independent set of columns of $K$.