We consider methods, both iterative and direct, for solving an R-linear system M z + M # z = b in C n with a pair of matrices M, M # ∈ C n×n and a vector b ∈ C n. Algorithms that avoid formulating the problem as an equivalent real linear system in R 2n are introduced. Conversely, this implies that real linear systems in R 2n can be solved with the methods proposed in this paper. Our study is motivated by Krylov subspace iterations with which using the real formulation can be disastrous in the standard linear case. Related matrix analysis and spectral theory are developed.
Real linear operators arise in a range of applications of mathematical physics. In this paper, basic properties of real linear operators are studied and their spectral theory is developed. Suitable extensions of classical operator theoretic concepts are introduced. Providing a concrete class, real linear multiplication operators are investigated and, motivated by the Beltrami equation, related problems of unitary approximation are addressed. Mathematics Subject Classification (2010). Primary 47A10; Secondary 47B38.
Abstract. The R-linear Beltrami equation appears in applications, such as the inverse problem of recovering the electrical conductivity distribution in the plane. In this paper, a new way to discretize the R-linear Beltrami equation is considered. This gives rise to large and dense R-linear systems of equations with structure. For their iterative solution, norm minimizing Krylov subspace methods are devised. In the numerical experiments, these improvements combined are shown to lead to speed-ups of almost two orders of magnitude in the electrical conductivity problem.
Linear algebra of factoring a matrix into the product of two matrices with special properties is developed. This is accomplished in terms of the so-called inverse of a matrix subspace which yields an extended notion for the invertibility of a matrix. The product of two matrix subspaces gives rise to a natural generalization of the concept of matrix subspace. Extensions of these ideas are outlined. Several examples on factoring are presented. (2000): 15A23, 65F30.
AMS subject classificationKey words: matrix factorization, inverse of a matrix subspace, product of matrix subspaces.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.