A generalization of the rotation matrix and related results
Let z be an independent complex variable. The author shows that applying the exponential map to z times the \(n\times n\) circulant matrix (skew circulant matrix), whose first row is 0,1,...,0, gives the circulant (skew circulant) matrix whose first row consists of the hyperbolic (trigonometric) functions of order n. This result provides a new generalization of the \(2\times 2\) rotation matrix, as well as a unified treatment of some properties of that generalization and of the matrices considered by \textit{A. Ungar} [Am. Math. Mon. 89, 688-691 (1982; Zbl 0529.33001)]. Infinitesimal generators and invariants of the corresponding Lie groups of transformations are obtained.
scientific article; zbMATH DE number 4010697