I am working through the book Fundamentals of Matrix Computations by David Watkins, and I ran into this one and it's stumping me. In my head, I understand the basic premise of it. However, I can't put pen to paper as to proving it. Proofs are not my strong point.
Let $M\in\Bbb{R}$ be positive definite, and let $R$ be the Cholesky factor of $M$, so that $M = R^TR$.
Show that $||M||_2 = ||R||_2^2$ and $\kappa_2(M) = \kappa_2(R)^2$.