if $A$ is a symmetric and nonnegative definite matrix then $A=LL^T$ for some $L$ lower triangular matrix.

Question

I need help with this problem:

Prove that if $A$ is a symmetric and nonnegative definite matrix then $A=LL^T$ for some $L$ lower triangular matrix. The terminology nonnegative definite means that $x^TAx\geq0$ for all $x$.

I have understood the answer for this case (2x2) Prove that if the matrix $\begin{bmatrix}a & b\\b & c\end{bmatrix}$ is nonnegative definite, then it has a factorization $LL^{T}$ but i would want how to extend to $\mathbb{R}^n$.

I'm grateful with any hint.

Answer 1

Try mathematical induction. Suppose $A_k=\pmatrix{A_{k-1}&v_k\\ v_k^T&a_k}$. If $A_{k-1}=L_{k-1}L_{k-1}^T$ for some lower triangular matrix $L_{k-1}$, try to prove that there exist a vector $x$ and a scalar $l$ such that $$ \pmatrix{A_{k-1}&v_k\\ v_k^T&a_k} =\pmatrix{L_{k-1}&0\\ x^T&l}\pmatrix{L_{k-1}^T&x\\ 0&l}. $$