On <i>k</i>-Circulant matrices involving the Lucas numbers of even index

Abstract Let k be a nonzero complex number. In this paper, we consider a k -circulant matrix whose first row is ( L 2 , L 4 , … , L 2 n ), where L n is the n th Lucas number. The formulae for the eigenvalues of such matrix are obtained. Namely, the result which can be obtained from Theorem 2 of [Bueno, A.C.F.: On r-circulant matrices with Fibonacci and Lucas numbers having arithmetic indices , AIP Conf. Proc. 1905 (2017), 030010] is improved because this result can not be applied in some cases. Since Theorem 4 in the same work relies on Theorem 2, the obtained formulae for the eigenvalues of a k -circulant matrix involving the Lucas numbers of even index show that there are cases when the result of Theorem 4 can also not be applied. Additionally, we present a more precise formula for the Euclidean norm of such matrix compared to the formula presented in Theorem 5 in the aforementioned reference. The upper and lower bounds for the spectral norm of a k -circulant matrix whose first row is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo stretchy="false">(</m:mo> <m:msubsup> <m:mi>L</m:mi> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mo>,</m:mo> <m:msubsup> <m:mi>L</m:mi> <m:mrow> <m:mn>4</m:mn> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mo>,</m:mo> <m:mo>…</m:mo> <m:mo>,</m:mo> <m:msubsup> <m:mi>L</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mi>n</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mo stretchy="false">)</m:mo> </m:math> $(L_{2}^{-1},L_{4}^{-1},\dots,L_{2n}^{-1})$ are also investigated. The obtained results are illustrated by examples.