1. The Classical Lanczos Method

At the time of Lanczos's work on the eigenvalue problem during the Second World War, most methods focused on finding the characteristic polynomial of matrices in order to find their eigenvalues. In fact, Lanczos's original paper [2] was also mostly concerned with this problem; however, he was trying to reduce the round-off errors in such calculations. He called his method the method of minimized iterations, which we will now review to lay the foundation.