On r-Circulant Matrices with Higher-Order Fibonacci Numbers

In this paper, we introduce and investigate a new class of r-circulant matrices whose entries are generated by higher-order Fibonacci numbers. Explicit representations of the eigenvalues of these matrices are derived by means of the Binet formula together with the structural properties of r-circulant matrices. Based on these representations, a closed-form expression for the determinant is obtained. In addition, several summation identities involving higher-order Fibonacci numbers are established, including formulas for partial sums, sums of squares, and weighted sums. These identities play a fundamental role in the derivation of the norm expressions and spectral estimates of the matrices. Furthermore, several matrix norms, including the Euclidean (Frobenius) norm, the 1-norm, the ∞-norm, and the spectral norm, are investigated in detail. Lower and upper bounds for the spectral norm are obtained for both cases |r|≥1 and |r|<1 by employing Hadamard product techniques and classical norm inequalities. Finally, numerical examples are presented to illustrate and validate the theoretical results.