# Computation of Cholesky decomposition of Gram matrix from its components

Let's assume I have a tall matrix $$\mathbf{X} \in \mathbb{C}^{m\times n}$$, where $$m \gg n$$. I form the Gram matrix $$\mathbf{A} = \mathbf{X}^*\mathbf{X}$$, where $$\mathbf{A} \in \mathbb{C}^{n\times n}$$ is Hermitian. As $$\mathbf{A}$$ is Hermitian, there exists a lower triangular matrix $$\mathbf{L} \in \mathbb{C}^{n\times n}$$ such that $$\mathbf{A} = \mathbf{L}\mathbf{L}^*$$ (where $$\mathbf{L}$$ is the Cholesky factor of $$\mathbf{A}$$). Is there a way to compute $$\mathbf{L}$$ without forming $$\mathbf{A}$$ first?

I saw some things based on Lyapunov equation solvers, but I can't figure how this could help here.

Yes. Run 'thin' QR factorization on $$X$$ and set $$L:=R^*$$. This means $$R$$ is square and $$Q$$ is tall and skinny, such that $$Q^*Q = I_n$$. To confirm: $$X=QR\implies X^*X =R^*Q^*QR = R^*R =LL^*$$