Solution Methods for $\mathbb R$-Linear Problems in $\mathbb C^n $

• Authors: Timo Eirola;Marko Huhtanen;Jan von Pfaler
• Venue: SIAM Journal on Matrix Analysis and Applications
• Year: 2003

We consider methods, both direct and iterative, for solving an ${\mathbb R}$-linear system $Mz+ M_{\#}\overline{z}= b$ in ${\mathbb C}^n$ with a pair of matrices $M, M_{\#} \in {\mathbb C}^{n \times n}$ and a vector $b\in {\mathbb C}^n$. Algorithms that avoid formulating the problem as an equivalent real linear system in ${\mathbb R}^{2n}$ are introduced. Conversely, this implies that real linear systems in ${\mathbb R}^{2n}$ can be solved with the methods proposed in this paper. Our study is motivated by Krylov subspace iterations, which when used with the real formulation can be disastrous in the standard linear case. Related matrix analysis and spectral theory are developed.