The Lorentz reciprocity theorem is simply a reflection of the fact that the linear operator O ^ ^ relating J and E at a fixed frequency ω ω (in linear media): J = O ^ ^ ⁡ ⁡ E where O ^ ^ ⁡ ⁡ E ≡ ≡ 1 i ω ω [ 1 μ μ ( ∇ ∇ × × ∇ ∇ × × ) − − ω ω 2 ε ε ] E is usually a symmetric operator under the " inner product " ( F , G ) = ∫ ∫ F ⋅ ⋅ G d V for vector fields F and G . (Technically, this unconjugated form is not a true inner product because it is not real-valued for complex-valued fields, but that is not a problem here. In this sense, the operator is not truly Hermitian but is rather complex-symmetric.) This is true whenever the permittivity ε and the magnetic permeability μ, at the given ω, are symmetric 3×3 matrices (symmetric rank-2 tensors) – this includes the common case where they are scalars (for isotropic media), of course. They need not be real – complex values correspond to materials with losses, such as conductors with finite conductivity σ (which is included in ε via ε ε → → ε ε + i σ σ / ω ω ) – and because of this, the reciprocity theorem does not require time reversal invariance. The condition of symmetric ε and μ matrices is almost always satisfied; see below for an exception.