On k-circulant matrices with the Mersenne numbers having arithmetic indices

Abstract

Let k be a nonzero complex number. In this paper, we consider a k-circulant matrix whose first row is (M_{s}, M_{s+t}, M_{s+2t},..., M_{s+(n-2)t}, M_{s+(n-1)t}), where M_{n} is the n^{th} Mersenne number, s is a non-negative integer and t is a positive integer. The formulae for the eigenvalues of such matrix are obtained. That formulae improve the result of Theorem 2.3. [15] (because of the result of Theorem 2.3. [15] can not be applied in some cases) and show that there are cases when the result of Theorem 2.9. [15] can also not be applied. Then, we considerthe norms of such matrix. Namely, the obtained formulae for the 1-norm, the \infty-\norm, the Euclidean norm and the spectral norm of such matrix extend (and correct) the results of, respectively, Theorem 3.3. [15], Theorem 3.4. [15] and Theorem 3.6. [15]. At the end of the paper, we also obtain the bounds for the spectral norm of a $k$-circulant matrix whose first rowis (M_{s}^{-1}, M_{s+t}^{-1}, M_{s+2t}^{-1},..., M_{s+(n-2)t}^{-1}, M_{s+(n-1)t}^{-1}) provided that s is a positive integer.