On ideal matrices whose entries are the generalized $k-$Horadam numbers

Ideal matrices, which generalize circulant and $r-$circulant matrices, play a key role in Ajtai's construction of collision-resistant hash functions. In this paper, we study ideal matrices whose entries are the generalized $k-$Horadam numbers, which represent a generalization of second-order sequences and include many well-known sequences such as Fibonacci, Lucas, and Pell numbers as special cases. We derive two explicit formulas for calculating the eigenvalues and determinants of these matrices. Additionally, we obtain upper bounds for the spectral norm and the Frobenius norm of ideal matrices with generalized $k-$Horadam number entries. These results not only extend existing findings on ideal matrices but also highlight the versatility and applicability of generalized $k-$Horadam numbers in matrix theory and related fields.