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

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

Effective Methods of QR-Decompositions of Square Complex Matrices by Fast Discrete Signal-Induced Heap Transforms

The purpose of this work is to present an effective tool for computing different QR-decompositions of a complex nonsingular square matrix. The concept of the discrete signal-induced heap transform (DsiHT, Grigoryan 2006) is used. This transform is fast, has a unique algorithm for any length of the input vector/signal and can be used with different complex basic 2x2 transforms. The DsiHT zeroes all components of the input signal while moving or heaping the energy of the signal into one component, such as the first. We describe three different types of QR-decompositions that use the basic transforms with the T, G, and M-type complex matrices we introduce, and also without matrices, but using analytical formulas. We also present the mixed QR-decomposition, when different type DsiHTs are used at different stages of the algorithm. The number of such decompositions is greater than 3^((N-1)), for an NxN complex matrix. Examples of the QR-decomposition are described in detail for the 4x4 and 6x6 complex matrices and compared with the known method of Householder transforms. The precision of the QR-decompositions of NxN matrices, when N are 6, 13, 17, 19, 21, 40, 64, 100, 128, 201, 256, and 400 is also compared. The MATLAB-based scripts of the codes for QR-decompositions by the described DsiHTs are given.Comment: 19 pages, 4 figures, 1 tabl