Transformation from Real Homogeneous Systems of Degree ℓ to Complex Homogeneous Systems of Degree(ℓ,0)

In eigenvalue analysis, transformation from real systems to complex systems is very important. First, we clarify a necessary and sufficient condition that solutions of real nonlinear systems coincide with solutions of transformed complex nonlinear systems in the real subspace. Moreover, we propose a complex transformation such that a) real homogeneous systems of degree ℓ with respect to r are transformed to complex homogeneous systems of degree (ℓ,0) with respect to r and b) solutions of real systems coincide with solutions of transformed complex systems in the real subspace. Then, we show examples.