Abstract: By means of the properties of Hermitian generalized anti-Hamiltonian semi-definite matrices, the authors give some necessary and sufficient conditions for the solvability of the linear matrix equation in Hermitian generalized anti-Hamiltonian semi-definite matrix set, and obtain a general expression of the solution for a solvable case. The authors also
give some necessary and sufficient conditions of the solvability for inverse enginvalume problem of Hermitian generalized anti-Hamiltonian semi-definite matrices. For any n× n complex matrix, the authors derive the representation of its unique optimal approximation.

Key words: Hermitian generalized anti-Hamiltonian semi-definite matrices, Linear matrix equations, Inverse enginvalume problems, Optimal approximation