Wilde, Alan C. Algebras of operators isomorphic to the circulant algebra. (English) Zbl 0672.15004 Proc. Am. Math. Soc. 105, No. 4, 808-816 (1989). The algebra of \(n\times n\) circulant matrices has a specific structure. The paper displays different operators on linear vector spaces that have the same structure, i.e. are isomorphic. A cyclic group of automorphisms on the circulant algebra generalizing conjugation is introduced. The group ring over C of this group is isomorphic to that of circulants themselves. Functional equations, whose solutions are functions \(C^ n\to C\), are solved using cyclic and idempotent linear operators on the space of functions \(C^ n\to C:\) this algebra of linear operators is isomorphic to \(n\times n\) circulants. Finally, one proves that the group ring of a linear involution on the space V of functions on \(n\times n\) complex circulants is isomorphic to \(2\times 2\) complex circulant matrices. Reviewer: R.Covaci Cited in 2 Documents MSC: 15A30 Algebraic systems of matrices 15A27 Commutativity of matrices Keywords:circulant matrices; linear vector spaces; cyclic group of automorphisms; circulant algebra; group ring; linear involution PDFBibTeX XMLCite \textit{A. C. Wilde}, Proc. Am. Math. Soc. 105, No. 4, 808--816 (1989; Zbl 0672.15004) Full Text: DOI References: [1] Philip J. Davis, Circulant matrices, John Wiley & Sons, New York-Chichester-Brisbane, 1979. A Wiley-Interscience Publication; Pure and Applied Mathematics. · Zbl 0418.15017 [2] Alan C. Wilde, Solutions of equations containing primitive roots of unity, J. Undergraduate Math., 3 (1971), 25-28. [3] Thomas Muir, A treatise on the theory of determinants, Revised and enlarged by William H. Metzler, Dover Publications, Inc., New York, 1960. [4] Kenneth R. Leisenring, The bicomplex plane (U-M manuscript, submitted for publication). · Zbl 0048.37204 [5] Alan C. Wilde, Cauchy-Riemann conditions for algebras isomorphic to the circulant algebra, J. Univ. Kuwait Sci. 14 (1987), no. 2, 189 – 204 (English, with Arabic summary). · Zbl 0638.35016 [6] Alan C. Wilde, Differential equations involving circulant matrices, Rocky Mountain J. Math. 13 (1983), no. 1, 1 – 13. · Zbl 0516.34016 · doi:10.1216/RMJ-1983-13-1-1 [7] Alan C. Wilde, Commutative projection operators, Atti Sem. Mat. Fis. Univ. Modena 35 (1987), no. 1, 167 – 172. · Zbl 0641.15013 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.