The QRD and SVD of matrices over a real algebra

2015/12/07 by Ginzberg, Paul, Mavroyiakoumou, Christiana
#15A18 #15A33 #15A66 #16S34 #16S35 #65F25 #65F30 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1512.02121

Abstract

Recent work in the field of signal processing has shown that the singular value decomposition of a matrix with entries in certain real algebras can be a powerful tool. In this article we show how to generalise the QR decomposition and SVD to a wide class of real algebras, including all finite-dimensional semi-simple algebras, (twisted) group algebras and Clifford algebras. Two approaches are described for computing the QRD/SVD: one Jacobi method with a generalised Givens rotation, and one based on the Artin-Wedderburn theorem.

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