Find necessary and sufficient conditions on $a,b,c$ such that the matrix $\begin{bmatrix}a & b\\b & c\end{bmatrix}$ has a factorization $LL^{T}$

Question

Find necessary and sufficient conditions on $a,b,c$ such that the matrix $$\begin{bmatrix}a & b\\b & c\end{bmatrix}$$ has a factorization $LL^{T}$ in which $L$ is lower triangular.**

I am using the following to demonstrate this:

• Prove that if the matrix $\begin{bmatrix}a & b\\b & c\end{bmatrix}$ is nonnegative definite, then it has a factorization $LL^{T}$

• Find the precise conditions on $a,b,c$ so that $\left[\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right]$ is nonnegative definite.

With the above, can I conclude that the necessary and sufficient conditions must be the following?

$$a \geq 0, \quad c \geq 0, \quad b^2 \leq ac$$

Thank you very much.

Answer 1

Yes, those conditions are sufficient and necessary.

A remark is that $$a \ge 0, b^2 \le ac \implies c \ge 0$$