Linear maps transforming H-unitary matrices

Let H1be an n × n invertible Hermitian matrix, and let U(H1) be the group of n × n H1-unitary matrices, i.e., matrices A satisfying A*H1A = H1. Suppose H2is an m × m invertible Hermitian matrix. We show that a linear transformation φ: Mn→ Mmsatisfies φ(U(H1))⊂U(H2) if and only if there exist invertible matrices S ∈ Mm, U,V ∈ U(H2) such that S* H2S = [(Ia⊕ - Ib) ⊗ H1] ⊕[(Ic⊕-Id) ⊗ (H1-1)t],and φ has the form A → US[(Ia+b⊗ A) ⊕ (Ic+d⊗ At)]S-1V, where a, b, c and d are nonnegative integers satisfying (a+b+c+d)n = m. Assume H1has inertia (p,q) and H2has inertia (r,s). Then there is a linear transformation mapping U(H1) into U(H2) if and only if there are nonnegative integers u and v such that (r, s) = u(p,q)+v(q,p). These results generalize those of Marcus, Cheung and Li.