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    On the spectral norm of r-circulant matrices with the Pell and Pell-Lucas numbers
    (SPRINGER INTERNATIONAL PUBLISHING AG, 2016) Turkmen, Ramazan; Gokbas, Hasan
    Let us define A = C-r (a(0), a(1),..., a(n-1)) to be a nxn r-circulant matrix. The entries in the first row of A = C-r (a(0), a(1),..., a(n-1)) are a(i) = P-i, a(i) = Q(i), a(i) = P-i(2) or a(i) = Q(i)(2) (i = 0, 1, 2,..., n - 1), where P-i and Q(i) are the ith Pell and Pell-Lucas numbers, respectively. We find some bounds estimation of the spectral norm for r-Circulant matrices with Pell and Pell-Lucas numbers.
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    Some norm inequalities for special Gram matrices
    (DE GRUYTER OPEN LTD, 2016) Turkmen, Ramazan; Kan, Osman; Gokbas, Hasan
    In this paper we firstly give majorization relations between the vectors F-n = {f(0), f(1),..., f(n-1)}, L-n = {l(0), l(1),..., l(n-1)} and P-n = {p(0), p(1),..., p(n-1) g which constructed with fibonacci, lucas and pell numbers. Then we give upper and lower bounds for determinants, Euclidean norms and Spectral norms of Gram matrices G(F) = < F-n, F-n >, G(L) = < L-n, L-n >, G(P) = < P-n, P-n >, G(FL) = < F-n, L-n >, G(FP) = < F-n, P-n >.