Least squares solutions to the rank-constrained matrix approximation problem in the Frobenius norm

In this paper, we discuss the following rank-constrained matrix approximation problem in the Frobenius norm: ‖C-AX‖=min\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert C-AX\Vert =\min $$\end{document} subject to rkC1-A1X=b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \text{ rk }\left( {C_1 - A_1 X} \right) = b $$\end{document}, where b is an appropriate chosen nonnegative integer. We solve the problem by applying the classical rank-constrained matrix approximation, the singular value decomposition, the quotient singular value decomposition and generalized inverses, and get two general forms of the least squares solutions.