A classical algorithm for eigenvalues and eigenvectors calculation

The following is an algorithm for the calculation of the eigenvalues of a square matrix. It is present in almost all the "classic" texts of numerical analysis. It is an indirect method and is an ill-posed problem as opposed to the modern ones (for example, QR). I, however, did not find the theory that allows the calculation of the eigenvectors, at the end of last century, I developed the algorithm, shown below. The accuracy is to be verified.