On the Relationship Between Real and Complex Linear Systems

2017/05/26 by Cédric Josz, Josz, Cédric
Computer Science · #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1706.00268

openalex publication_date 2017/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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.

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