HAMILTON AND JACOBI MEET AGAIN - QUATERNIONS AND THE EIGENVALUE PROBLEM

The algebra isomorphism between M(4)(R) and HxH, where H is the algebr a of quaternions, has unexpected computational payoff: it helps constr uct an orthogonal similarity that 2x2 block-diagonalizes a 4x4 symmetr ic matrix. Replacing plane rotations with these more powerful 4x4 rota tions leads to a quaternion-Jacobi method in which the ''weight'' of f our elements (in a 2x2 block) is transferred all at once onto the diag onal. Quadratic convergence sets in sooner, and the new method require s at least one fewer sweep than plane-Jacobi methods. An analogue of t he sorting angle for plane rotations is developed for these 4x4 rotati ons.