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Least squares solutions to the rank-constrained matrix approximation problem in the Frobenius norm. (English) Zbl 1428.15007

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

MSC:

15A12 Conditioning of matrices
15A24 Matrix equations and identities
15A09 Theory of matrix inversion and generalized inverses
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
65F35 Numerical computation of matrix norms, conditioning, scaling
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