2. QR algorithm for the non-symmetrical case

In this paragraph, we assume that the matrix A, whose eigenvalues we are looking for, is real and that we keep the calculations real for as long as possible. This is indeed the most common case. Although the use of complex arithmetic in the case of non-symmetric matrices is conceptually simpler than the restriction to real calculations, since it eliminates the need to deal with special cases, from a computing point of view, we prefer to avoid complex calculations as they generally slow down execution. However, all the calculations described can be performed in complex arithmetic, taking care to use the Hermitian scalar product of two complex vectors x and y defined by