Quaternion Singular Value Decomposition based on Bidiagonalization to a Real Matrix using Quaternion Householder Transformations

Details Quaternion Singular Value Decomposition based on Bidiagonalization to a Real Matrix using Quaternion Householder Transformations Quaternion Singular Value Decomposition based on Bidiagonalization to a Real Matrix using Quaternion Householder Transformations

2006-03-10

arxiv.org/abs/math/0603251v1

Applied Mathematics and Computation, 182 (1), 1 November 2006, 727-738

Mathematics

Numerical Analysis

Scientific paper

10.1016/j.amc.2006.04.032

We present a practical and efficient means to compute the singular value decomposition (svd) of a quaternion matrix A based on bidiagonalization of A to a real bidiagonal matrix B using quaternionic Householder transformations. Computation of the svd of B using an existing subroutine library such as lapack provides the singular values of A. The singular vectors of A are obtained trivially from the product of the Householder transformations and the real singular vectors of B. We show in the paper that left and right quaternionic Householder transformations are different because of the noncommutative multiplication of quaternions and we present formulae for computing the Householder vector and matrix in each case.

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