Minimum rank (skew) Hermitian solutions to the matrix approximation problem in the spectral norm

In this paper, we discuss the following two minimum rank matrix approximation problems in the spectral norm: (i)For given A=AH∈Cm×m,B∈Cm×n, determining X∈S1X∈S1, such that rank(X)=minY∈S1rank(Y),S1={Y=YH∈Cn×n:‖A−BYBH‖2=min}S1={Y=YH∈Cn×n:‖A−BYBH‖2=min}.(ii)For given A=−AH∈Cm×m,B∈Cm×n, determining X∈S2X∈S2, such that rank(X)=minY∈S2rank(Y),S2={Y=−YH∈Cn×n:‖A−BYBH‖2=min}S2={Y=−YH∈Cn×n:‖A−BYBH‖2=min}.By applying the norm-preserving dilation theorem, the Hermitian-type (skew-Hermitian-type) generalized singular value decomposition (HGSVD, SHGSVD), we characterize the expressions of the minimum rank and derive a general form of minimum rank (skew) Hermitian solutions to the matrix approximation problem.