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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.

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