It's often stated (eg: in Numerical Recipes in C) that Cholesky factorization is numerically stable even without column pivoting, unlike LU decomposition, which usually need pivoting schemes.
But I've never seen a proof or argument for why that's true. Does it have to do with the matrix class it works on being Positive Definite? If you LU decompose Positive Definite matrices without pivoting is it still numerically unstable?