#### Vol. 10, No. 1, 2017

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A generalization of the matrix transpose map and its relationship to the twist of the polynomial ring by an automorphism

### Andrew McGinnis and Michaela Vancliff

Vol. 10 (2017), No. 1, 43–50
##### Abstract

A generalization of the notion of symmetric matrix was introduced by Cassidy and Vancliff in 2010 and used by them in a construction that produces quadratic regular algebras of finite global dimension that are generalizations of graded Clifford algebras. In this article, we further their ideas by introducing a generalization of the matrix transpose map and use it to generalize the notion of skew-symmetric matrix. With these definitions, an analogue of the result that every $n×n$ matrix is a sum of a symmetric matrix and a skew-symmetric matrix holds. We also prove an analogue of the result that the transpose map is an antiautomorphism of the algebra of $n×n$ matrices, and show that the antiautomorphism property of our generalized transpose map is related to the notion of twisting the polynomial ring on $n$ variables by an automorphism.

##### Keywords
transpose, automorphism, symmetric, skew-symmetric, polynomial ring, twist
##### Mathematical Subject Classification 2010
Primary: 15A15, 15B57, 16S50, 16S36