  We consider the problem of solving a linear system of equations which
involves complex variables and their conjugates. We characterize when it
reduces to a complex linear system, that is, a system involving only complex
variables (and not their conjugates). In that case, we show how to construct
the complex linear system. Interestingly, this provides a new insight on the
relationship between real and complex linear systems. In particular, any real
symmetric linear system of equations can be solved via a complex linear system
of equations. Numerical illustrations are provided. The mathematics in this
manuscript constitute an exciting interplay between Schur's complement,
Cholesky's factorization, and Cauchy's interlace theorem.
