Let $A$ be symmetric and positive definite. Suppose $A=LL^T$ is its Cholesky decomposition. Prove that $||A||_2=||L||_2^2$.

Question

Let $A$ be symmetric and positive definite. Suppose $A=LL^T$ is its Cholesky decomposition. Prove that $||A||_2=||L||_2^2$.

This is an exercise in my Numerical Analysis book. The offcial hint to this problem says:

$||L||_2^2=\rho(L^TL)$

Following the hint, I can only show that $||L^T||_2^2=\rho(LL^T)=\rho(A)=||A||_2$. If the original proposition is true, then it is necessary $||L||_2=||L^T||_2$ for all nonsingular lower triangular $L$. But I don't think this is always true. Is there a typo in the problem?