Hamilton and Jacobi Meet Again: Quaternions and the Eigenvalue Problem
Niloufer Mackey · SIAM Journal on Matrix Analysis and Applications · 1995
The algebra isomorphism between $\mathcal{M}_4 ( \mathcal{R} )$ and $\mathcal{H} \otimes \mathcal{H}$, where $\mathcal{H}$ 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.