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Algebras of operators isomorphic to the circulant algebra. (English) Zbl 0672.15004

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

MSC:

15A30 Algebraic systems of matrices
15A27 Commutativity of matrices
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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
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