So the Cholesky decomposition theorem states that that any real symmetric positive-definite matrix $M$ has a Cholesky decomposition $M= LL^\top$ where $L$ is a lower triangular matrix.
Given $M$, we already know there are fast algorithms to calculate its Cholesky factor $L$.
Now, suppose I was given a rectangular $m\times n$ matrix $A$, and I knew that $A^\top A$ was positive definite. Is there a way to calculate the Cholesky factor $L$ of $A^\top A$ without computing $A^\top A$ explicitly and then applying Cholesky factorization algorithms?
If $A$ is a very large rectangular matrix performing $A^\top A$ explicitly seems very expensive and hence the question.