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Minimum rank positive semidefinite solution to the matrix approximation problem in the spectral norm. (English) Zbl 1442.15037

Summary: In this paper, we discuss the following minimum rank matrix approximation problem in the spectral norm: \[ \underset{X \geqslant 0}{\min}\ r(X) \quad \text{subject to} \quad \| A - B X B^\ast \|_2 = \min, \] where \(A \in \mathbb{C}_\geqslant^{m \times m}\) and \(B \in \mathbb{C}^{m \times n}\). By using the positive-semidefinite-type generalized singular value decomposition, we derive the expressions of the minimum rank and the minimum rank positive semidefinite solution to the above matrix approximation problem.

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A03 Vector spaces, linear dependence, rank, lineability
15A45 Miscellaneous inequalities involving matrices
65F55 Numerical methods for low-rank matrix approximation; matrix compression

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References:

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