Quaternion singular value decomposition based on bidiagonalization to a real or complex matrix using quaternion Householder transformations

Abstract

We present a practical and efficient means to compute the singular value decomposition (SVD) of a real or complex quaternion matrix A based on bidiagonalization of A to a real or complex 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 or complex singular vectors of B. We show in the paper that left and right quaternionic Householder transformations are different because of the non-commutative multiplication of quaternions and we present formulae for computing the Householder vector and matrix in each case.