Equality constrained least squares problem over quaternion field
type: Item
instance of: scholarly article
title: Equality constrained least squares problem over quaternion field
zbMATH Open document ID: 1048.65036
DOI: 10.1016/S0893-9659(03)90012-7
author: Musheng Wei
author: Tongsong Jiang
published in: Applied Mathematics Letters
publication date: 2004-06-11T00:00:00Z
review text: Given a quaternion \(\gamma = x_0+x_1 i+x_2 j+x_3 k = \alpha +\beta j\in{\mathbb Q}\), where \(x_i\in {\mathbb R}\), \(i^2=j^2=k^2=-1\) and \(ij=-ji=k\), and a quaternion matrix \(A=(a_{ij})\in {\mathbb Q}^{m\times n}\), the complex representation of \(\gamma\) and \(A\) are \[ \gamma^\sigma =\begin{pmatrix} x_0+x_1\sqrt{-1} & x_2+x_3\sqrt{-1} \\ -x_2+x_3\sqrt{-1} & x_0-x_1\sqrt{-1}\end{pmatrix} = \begin{pmatrix} \alpha & \beta \\ -\overline\beta & \overline\alpha \end{pmatrix}\in {\mathbb C}^{2\times 2} \] and \[ \sigma(A)=A^\sigma=((a_{ij})^\sigma)=\left(\begin{pmatrix} \alpha_{ij} & \beta_{ij} \\\;-\overline\beta_{ij} & \overline\alpha_{ij} \end{pmatrix}\right)\in{\mathbb C}^{2m\times 2n}. \] The authors use the above complex representation and the generalized singular value decomposition of quaternion matrices to define the \(\ell^2\) and Frobenius norms of quaternion matrices and to give necessary and sufficient conditions to solve the quaternion least square problem (LSE) in the Frobenius norm: \(\| Ax-b\| =\min\), subject to \(Bx=d\), where \(A\in {\mathbb Q}_r^{m\times n}\) (here \(m\geq n\) and the subscript \(r\) is the rank of \(A\)), \(B\in {\mathbb Q}_s^{p\times n}\), \(b\in {\mathbb Q}^{m\times 1}\) and \(d\in {\mathbb Q}^{m\times 1}\). A practical computing procedure to solve the LSE problem is given.
reviewed by: Rémi Vaillancourt
Mathematics Subject Classification ID: 65F20
Mathematics Subject Classification ID: 15B33
zbMATH DE Number: 2073700
zbMATH Keywords: equality constrained least squares problems
zbMATH Keywords: quaternion matrices
zbMATH Keywords: complex representation
zbMATH Keywords: generalized singular value decomposition
zbMATH Keywords: Frobenius norms
MaRDI profile type: Publication
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full work available at URL: s0893-9659(03)90012-7
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description: scientific article; zbMATH DE number 2073700