# Proving matrix factorization using Schur complement

In my studies of matrix analysis I came across this problem:

Let $A$ be a real symmetric matrix which is also positive semidefinite and we are asked to prove, by induction on the dimension of $A$ that there exist the following factorization $A=LL^T$ and $A=UU^T$ where $L,U$ are lower and upper triangular matrices, respectively. We are instructed to prove this fact using the Schur complement, which is defined for the following block matrix $A = \left( \begin{array}{ccc} B & C \\ C^T & E \end{array} \right)$ where $E$ is principal square submatrix of $A$ the Schur complement is defined as $A/E = B-CE^{\dagger}C^T$ where $E^{\dagger}$ is the Moore-Penrose generalized inverse.

To be honest I have no idea how to prove this factorization using the assumption A is positive semidefinite and I have no idea how to carry out the induction using the Schur complement. I appreciate all help on this.