Abstract
In this paper, we obtain explicit Euclidean norm, eigenvalues, spectral norm and determinant of circulant matrix with the generalized Tribonacci (generalized (r, s, t)) numbers. We also present the sum of entries, the maximum column sum matrix norm and the maximum row sum matrix norm of this circulant matrix. Moreover, we give some bounds for the spectral norms of Kronecker and Hadamard products of circulant matrices of (r, s, t) and Lucas (r, s, t) numbers.
Highlights
By using Binet’s formula of generalized Tribonacci numbers and the following identities α + β + γ = r, αβ + αγ + βγ = −s, αβγ = t, we obtain n−1 xk Wk n−1 xk p1αk p3γk (α − β)(α − γ) (β − α)(β − γ) (γ − α)(γ − β) p1n − 1 +
We present the sum of entries, the maximum column sum matrix norm and the maximum row sum matrix norm of this circulant matrix
We give some bounds for the spectral norms of Kronecker and Hadamard products of circulant matrices of (r, s, t) and Lucas (r, s, t) numbers
Summary
The generalized (r, s, t) sequence (or generalized Tribonacci sequence or generalized 3-step Fibonacci sequence). Keywords and phrases: (r, s, t) numbers, circulant matrix, Tribonacci numbers, norm, determinant, eigenvalues This sequence has been studied by many authors, see for example [2, 3, 4, 10, 11, 18, 22, 28, 30, 44, 41, 45, 51, 52]. (r, s, t) sequence{Gn}n≥0 and Lucas (r, s, t) sequence {Hn}n≥0 are defined, respectively, by the third-order recurrence relations. The following Theorem presents a summing formula of generalized Tribonacci numbers with positive subscripts. By using Binet’s formula of generalized Tribonacci numbers and the following identities α + β + γ = r, αβ + αγ + βγ = −s, αβγ = t, we obtain n−1 xk Wk n−1 xk p1αk p2βk p3γk (α − β)(α − γ) (β − α)(β − γ) (γ − α)(γ − β) k=0 k=0. The case x = 1 of the above theorem can be given as follows. It is given in Soykan [42, Theorem 2.1]. Using Theorems 1.1, 1.2 and 1.3, we give relatively short proofs for our results which are given
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