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

Thaniporn Chaysri, Nevena Jakovčević Stor and Ivan Slapničar ()
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Thaniporn Chaysri: Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Rudjera Boškovića 32, 21000 Split, Croatia
Nevena Jakovčević Stor: Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Rudjera Boškovića 32, 21000 Split, Croatia
Ivan Slapničar: Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Rudjera Boškovića 32, 21000 Split, Croatia

Mathematics, 2024, vol. 12, issue 9, 1-21

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 ( n 2 ) 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.

Keywords: eigenvalue decomposition; matrices of quaternions; arrowhead matrix; diagonal-plus-rank- k matrix (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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