Equality constrained least squares problem over quaternion field
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.
equality constrained least squares problems
generalized singular value decomposition
scientific article; zbMATH DE number 2073700
scientific article