On the Relationship Between Real and Complex Linear Systems
Provides theoretical insight into the relationship between real and complex linear systems, but the practical impact is limited as it is a theoretical characterization without empirical validation.
The paper characterizes when a linear system involving complex variables and their conjugates reduces to a complex linear system (without conjugates), and shows that any real symmetric linear system can be solved via a complex linear system. Numerical illustrations are provided.
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.