Abstract
In this paper, we use the algebra methods, the properties of the r-circulant matrix and the geometric circulant matrix to study the upper and lower bound estimate problems for the spectral norms of a geometric circulant matrix involving the generalized k-Horadam numbers, and we obtain some sharp estimations for them. We can also give a new estimation for the norms of a r-circulant matrix involving the generalized k-Horadam numbers.
Highlights
Let n ≥ 2 be an integer, r be any real or complex number
For the matrices A and B as mentioned above we have rn–1 rn–2 rn–3 · · · r 1 n×n and
The methods of this paper can be deduced on the norms of r-circulant matrix, circulant matrix, geometric circulant matrices in particular with all second-order sequences, leading to better estimations than in reference [18]
Summary
Let n ≥ 2 be an integer, r be any real or complex number. a n × n r-circulant matrix. In [8], Yazlik and Taskara have studied eigenvalues, determinant and the spectral norms of circulant matrix involving the generalized k-Horadam numbers. In [10], Can and Naim have defined geometric circulant matrices and studied the bounds for the spectral norms of geometric circulant matrices involving the generalized Fibonacci number and Lucas numbers. Considering the above articles, on the one hand, we obtain a new lower and upper bounds estimates for the spectral norms of the geometric circulant matrix with the generalized k-Horadam numbers, at the same time, we get the spectral norms of geometric circulant matrices involving all second-order recurrence sequences or polynomials. We get the bounds for the spectral norms of geometric circulant matrices involving the generalized Fibonacci number and Lucas numbers [10].
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