Linear maps transforming H-Unitary Matrices

Let H1 be 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 H2 is an m × m invertible Hermitian matrix. We show that a linear transformation φ : Mn → Mm satisfies φ(U(H1)) ⊆ U(H2) if and only if there exist invertible matrices S ∈ Mm, U, V ∈ U(H2) such that SH2S = [(Ia ⊕−Ib)⊗H1]⊕ [(Ic ⊕−Id)⊗ (H−1 1 )], and φ has the form A 7→ 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 H1 has inertia (p, q) and H2 has 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. AMS Subject Classifications 15A04, 15A57, 15A63.