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

Let k be a non-zero complex number. In this paper, we consider a k -circulant matrix whose first row is (M_s, Mₛ₊ₜ, Mₛ₊₂ₜ, , Mₛ₊₍ₙ₋₂₎ₜ, Mₛ₊₍ₙ₋₁₎ₜ) , where M_s is the nᵗʰ 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. [20] (because the result of Theorem 2.3. [20] can not be applied in some cases) and show that there are cases when the result of Theorem 2.9. [20] can also not be applied. Then, we consider the norms of such matrix. Namely, the obtained formulae for the 1-norm, the ∞ -norm, the Euclidean norm and the spectral norm of such matrix extend (and correct) the results of, respectively, Theorem 3.3. [20], Theorem 3.4. [20] and Theorem 3.6. [20]. At the end of the paper, we also obtain the bounds for the spectral norm of a k -circulant matrix whose first row is (M_s⁻¹, Mₛ₊ₜ⁻¹, Mₛ₊₂ₜ⁻¹, , Mₛ₊₍ₙ₋₂₎ₜ⁻¹, Mₛ₊₍ₙ₋₁₎ₜ⁻¹) provided that s is a positive integer.