1. The problem statement, all variables and given/known data Hi, I have a question: Prove that a square matrix of order n has an LU-factorization if and only if the origin is not contained in the numerical range. 2. Relevant equations The numerical range is found using the Euclidean inner scalar product <Ax,x> where <x,x>=1 and. The numerical range is A(W)=(<Ax,x> such that <x,x> =1 for all x reals) 3. The attempt at a solution I tried considering the 2x2 matrix case but got stuck.:) Thanks in advance!