Inverse Problems Of Anti-Hermitian Generalized Hamiltonian Matrix On A Linear Manifold

Let J∈R^n×n be a orthogonal anti-symmetric matrix, that is, JJ^T = J^TJ = I_n, J^T = -J. For an A∈C^n×n, if A^H = -A, JAJ = A^H, we say that A is an anti-Hermitian generalized Hamiltonian matrix. V is a linear space of dimension n on the number field F and P is called a linear manifold of V, if P - M + a = {m + a|m G M}, in which M is a subspace of V and a is a fixed vector in V; Meanwhile, we say that the dimension of P is the same to that of M. In this paper, we mainly discussed the inverse problems of anti-Hermitian generalized Hamiltonian matrix on a linear manifold.Firstly, we found the solution of the linear manifold S = {A∈AHHC^n×n|f₁(A) = ||AX₁ - B₁||² + ||Y₁A - B₂||² = min}. After that, we got the least square solution of the matrix equation f₂{A) = ||AX₂ - C₁||² + ||Y₂A - C₂||² = 0 in the linear manifold S. Then, we studied the necessary and sufficient conditions of the matrix equation f₂(A) = 0 when it is solvable in the linear manifold S. At the same time, we gave the expression of its solution. Furthermore, we talked a special situation of the above matrix equation when C₁ = X₂?₁, C₂ = ?₂Y₂, in which ?₁ = diag(λ₁₁,λ₁₂,…,λ_1k2)∈C^{k₂×k₂}, ?₂ = diag(λ₂₁,λ₂₂,…,λ_2l2)∈C^{l₂×l₂}, that is, it's the left and right inverse eigenvalue problem of the matrix equation. We also talked the matrix optimal approximation problem under the above-mentioned three situations. Finally, we gave the algorithms and numerical examples for these problems.