# Decompositions of matrices and linear transformations

Wang, Lu (2011) Decompositions of matrices and linear transformations. Masters thesis, Memorial University of Newfoundland. [English] PDF - Accepted Version Available under License - The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. Download (10MB)

## Abstract

The aim of this thesis is to discuss how to express a matrix (or a linear transformation) as the sum of two invertible matrices (or invertible linear transformations) with some constraints. The work for this thesis is two-fold. Firstly, it is proved that if R is a semilocal ring or an exchange ring with primitive factors Artinian then R satisfies the Goodearl-Menal condition if and only if no homomorphic images of R is isomorphic to either Z₂ or Z₃ or M₂ (Z₂). These results correct two existing results in the literature. Secondly, for the ring R of linear transformations of a right vector space over a division ring D, two results are proved in this thesis: (1) If |D| > 3, then for any a ∈ R there exists a unit u of R such that both a + u and a - u⁻¹ are units of R; (2) If |D| > 2, then for any a ∈ R there exists a unit u of R such that both a - u and a - u⁻¹ are units of R. Result (1) extends a result of H. Chen  that the ring of linear transformations of a countably generated right vector space over a division ring D with |D| > 3 satisfies the condition that for any a ∈ R, there exists u ∈ U (R) such that a + u and a - u⁻¹ ∈ U (R). And result (2) answers a question raised by H. Chen  whether the ring of linear transformations of a countable generated right vector space over a division ring D with |D| > 2 satisfies the condition that for any a ∈ R, there exists u ∈ U (R) such that a - u and a - u⁻¹ ∈ U (R). Connections of these conditions with some well-known conditions in ring theory are also discussed.

Item Type: Thesis (Masters) http://research.library.mun.ca/id/eprint/9890 9890 Includes bibliographical references (leaves 36-38). Science, Faculty of > Mathematics and Statistics 2011 Submission Matrices; Semilocal rings; Transformations (Mathematics); Decomposition (Mathematics)

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