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

Abstract

In this paper, we discuss the following rank-constrained matrix approximation problem in the Frobenius norm: $$\Vert C-AX\Vert =\min $$ subject to $$ \text{ rk }\left( {C_1 - A_1 X} \right) = b $$, 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.