On the Norms and Eigenvalues of $r$-Circulant Matrices with $k$-Mersenne and $k$-Mersenne–Lucas Numbers

In this work, we study the $r$-circulant matrix $ C_r = Circ_r(c_0, c_1,c_2,...,c_{n-1})$ such that the entries of $C_r $ are $c_i=M_{k,a+ib}$ or $c_i=R_{k,a+ib}$, where $M_{k,a+ib}$ and $R_{k,a+ib}$ are $k$-Mersenne and $k$-Mersenne–Lucas numbers, respectively. We obtain the eigenvalues and determinants for the matrices and some important identities for the $k$-Mersenne and $k$-Mersenne–Lucas numbers. Furthermore, we find norms and bounds estimation for the spectral norm for these $r$-circulant matrices.