The QR and QL Algorithms for Symmetric Matrices

The QR algorithm as developed by Francis [2] and Kublanovskaya [4] is conceptually related to the LR algorithm of Rutishauser [7]. It is based on the observation that if $$A = QR{\text{ and }}B{\text{ = }}RQ{\text{,}}$$ (1) where Q is unitary and R is upper-triangular then $$B = RQ = {Q^H}AQ,$$ (2) that is, B is unitarily similar to A. By repeated application of the above result a sequence of matrices which are unitarily similar to a given matrix A 1 may be derived from the relations $${A_s} = {Q_s}{R_s},{\rm{ }}{A_{s + 1}} = {R_s}{Q_s} = Q_s^H{A_s}{Q_s}$$ (3) and, in general, A s tends to upper-triangular form.