Hamilton and Jacobi Meet Again: Quaternions and the Eigenvalue Problem

The algebra isomorphism between M 4 (R) and HHH, where H is the algebra of quaternions, has unexpected computational payoo: it helps construct an orthogonal similarity that 22 block-diagonalizesa 44 symmetricmatrix. Replacing plane rotations with these more powerful 4 4 rotations leads to a quaternion-Jacobi method in which thèweight' of 4 elements (in a 2 2 block) is transferred all at once onto the diagonal. Quadratic convergence sets in sooner, and the new method requires at least one fewer sweep than plane-Jacobi methods. An analogue of the sorting angle for plane rotations is developed for these 4 4 rotations.