ON THE BOUNDS FOR THE NORMS OF R-CIRCULANT MATRICES WITH THE JACOBSTHAL AND JACOBSTHAL LUCAS NUMBERS

The Jacobsthal {jn}n∈N, and the Jacobsthal Lucas {cn}n∈N sequences are defined recurrently by jn = jn−1 + 2jn−2, j0 = 0, j1 = 1, n ≥ 2, (1) cn = cn−1 + 2cn−2, c0 = 2, c1 = 1, n ≥ 2, (2) respectively. There have been several papers on the norms of special matrices [7-10]. Solak [8] has defined A = [aij ] and B = [bij] as nxn circulant matrices, where aij = F(mod(j−i,n)) and bij = L(mod(j−i,n)) , then he has given some bounds for the A and B matrices concerned with the spectral and Eu-

There have been several papers on the norms of special matrices [7][8][9][10].Solak [8] has defined A = [a ij ] and B = [b ij ] as n × n circulant matrices, where a ij = F (mod(j−i,n)) and b ij = L (mod(j−i,n)) , then he has given some bounds for the A and B matrices concerned with the spectral and Euclidean norms.Shen and Cen [10] have given upper and lower bounds for the spectral norms of rcirculant matrices A = C r (F n−1 ).In addition, they also have obtained some bounds for the spectral norms of Hadamard and Kronecker products of these matrices.
In this paper we give lower and upper bounds for the spectral norms of the circulant matrices A = C(j 0 , j 1 , ..., j n−1 ) and B = C(c 0 , c 1 , ..., c n−1 ).
Recurrences (1) and (2) involve the characteristic equation Their Binet's formulas are defined by and and the spectral norm of matrix A is where Lemma 1.For any A, B ∈ M m,n (C) , the Hadamard product of A, B is entrywise product and defined by [5,6] and have the following properties C) be given, then the Kronecker product of A, B is defined by and have the following property [11] A 2 The sum formulas of the square of Jacobsthal and Jacobsthal-Lucas numbers Proposition 4. The summation of the squares of Jacobsthal sequence is written by using Jacobsthal numbers as the following: Proof.By using Binet forms we have Proposition 5.The summation of the squares of Jacobsthal numbers is written by using Jacobsthal-Lucas numbers as the following: Proof.From Binet forms we obtain Proposition 6.The summation of the squares of Jacobsthal-Lucas numbers is written by using Jacobsthal numbers as the following: Proof.By using Binet forms we have and for the other equality we have
Theorem 8. Let A = C(j 0 , j 1 , ..., j n−1 ) be circulant matrix, then we obtain 1 3 Proof.Let the matrix A is of the form above. From ( 5), ( 8), (10) we get On the other hand, let A = BoC where B, C are defined in (14) The proof is completed.
Theorem 9. Let the elements of the circulant matrix be Jacobsthal-Lucas numbers, A = C(c 0 , c 1 , ..., c n−1 ), then we obtain Proof.The matrix A is of the form  (17) 2 c n−2 c n−1 c 0 • • • c n−3