Fast Eigenvalue Decomposition of Arrowhead and Diagonal-Plus-Rank-k Matrices of Quaternions

Abstract

Quaternions are a non-commutative associative number system that extends complex numbers, first described by Hamilton in 1843. We present algorithms for solving the eigenvalue problem for arrowhead and DPRk (diagonal-plus-rank-k) matrices of quaternions. The algorithms use the Rayleigh Quotient Iteration with double shifts (RQIds), Wielandt’s deflation technique and the fact that each eigenvector can be computed in O(n) operations. The algorithms require O(n2) floating-point operations, n being the order of the matrix. The algorithms are backward stable in the standard sense and compare well to the standard QR method in terms of speed and accuracy. The algorithms are elegantly implemented in Julia, using its polymorphism feature.

Inverses and Determinants of Arrowhead and Diagonal-Plus-Rank-One Matrices over Associative Algebras

“…The determinants and the inverses of arrowhead and DPR1 matrices are computed using O(n) operations, just as the matrix-vector products above; this is unlike general matrices, where these functions require O(n 3 ) operations (see Sections 2 and 3 for details). This fact can be used in deriving fast algorithms for eigenvalue decomposition of such matrices [9]. The basic idea in deriving formulas in the noncommutative setting is that the operations within a particular formula should be executed in the specified order.…”