Abstract

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.

中文翻译：

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关于 $$r$$ 的范数和特征值 - 具有 $$k$$ 的循环矩阵 -Mersenne 和 $$k$$ -Mersenne–Lucas 数

摘要

在这项工作中，我们研究\(r\) -循环矩阵\( C_r = Circ_r(c_0, c_1,c_2,...,c_{n-1})\)使得\(C_r \)的条目是\(c_i=M_{k,a+ib}\)或\(c_i=R_{k,a+ib}\)，其中\(M_{k,a+ib}\)和\(R_{k ,a+ib}\)分别是\(k\) -Mersenne 和\(k\) -Mersenne–Lucas 数。我们获得了矩阵的特征值和行列式以及\(k\) -Mersenne 和\(k\) -Mersenne–Lucas 数的一些重要恒等式。此外，我们找到了这些\(r\)循环矩阵的谱范数的范数和界限估计。