# On the Relationship Between Real and Complex Linear Systems

## TL;DR

This paper explores the conditions under which complex linear systems involving conjugates can be simplified to standard complex linear systems, revealing new links between real and complex systems with practical numerical methods.

## Contribution

It characterizes when such systems reduce to complex linear systems and demonstrates how to construct these systems, offering new insights into the connection between real and complex linear algebra.

## Key findings

- Real symmetric systems can be solved via complex linear systems.
- Provides a method to construct complex systems from conjugate-involving systems.
- Numerical illustrations validate the theoretical results.

## Abstract

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.