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In this paper, we consider a linear constraint problem of Hermitian unitary symplectic matrices and its approximation. By constructing a simple unitary matrix  U , we verify that Hermitian unitary symplectic matrices are unitary similar to block diagonal Hermitian unitary matrices via  U , which simplifies and is crucial to solving the linear constraint problem, and is a special feature of this paper. Then, we solve the linear constraint problem completely, that is deriving the sufficient and necessary conditions of it and inducing Hermitian unitary symplectic solutions to it. We also obtain its optimal approximate solutions. Furthermore, the Procrustes problem of Hermitian unitary symplectic matrices is considered when the linear constraint problem has no solution.

Linear constraint problem of Hermitian unitary symplectic matrices