Hamilton and Jacobi Meet Again: Quaternions and the Eigenvalue Problem

The algebra isomorphism between $\cal{M}_{4}(\cal{R})$ and $\cal{H} \otimes \cal{H}$, where $\cal{H}$ is the algebra of quaternions, has unexpected computational payoff: it helps construct an orthogonal similarity that $2 \times 2$ block-diagonalizes a $4\times 4$ symmetric matrix. Replacing plane rotations with these more powerful $4 \times 4$ rotations leads to a quaternion-Jacobi method in which the "weight" of four elements (in a $2 \times 2$ 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 $4 \times 4$ rotations.