In this paper, we discuss the following problem: For given matrices A and B with A Hermitian, find a semi-positive definite matrix X with the minimal rank, subject to the spectral norm of A-BXB attains minimum. We propose two different methods to solve this problem. We characterize the expressions of the minimum rank and derive a general form of minimum rank positive semi-definite solutions to the matrix approximation problem in all these three cases. Three numerical examples are provided to illustrate our analysis.

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