On the norms of an r-circulant matrix with the generalized k-Horadam numbers

The r-circulant matrices [1‐3] have been one of the most interesting research areas in the field of computation mathematics. It is known that these matrices have a wide range of applications in signal processing, coding theory, image processing, digital image disposal, linear forecast and design of self-regress.

Although there are many works about special matrices and their norms, we can mainly depict the following ones since they will be needed for our results. In [4], Solak defined the n × n circulant matrices A = [ a i j ] and B = [ b i j ], where a i j ≡ F ( mod ( j − i , n ) ) and b i j ≡ L ( mod ( j − i , n ) ), and then investigated the upper and lower bounds for A and B, respectively. In [5], they defined circulant matrices involving k-Lucas and generalized k-Fibonacci numbers and also investigated the upper and lower bounds for the norms of these matrices. Later on, Shen and Cen in [6], found some other upper and lower bounds for the spectral norms of r-circulant matrices in the form A = C r ( F 0 , F 1 , … , F n − 1 ), B = C r ( L 0 , L 1 , … , L n − 1 ), where F n is the n th Fibonacci number and L n is the n th Lucas number. In [7], the authors found some other upper and lower bounds for the spectral norms of r-circulant matrices associated with the k-Fibonacci and k-Lucas numbers, respectively. It is clear that the same study about similar subject can also be done for different numbers. For instance, in [8], authors defined a circulant matrix whose entries are the generalized k-Horadam numbers and then computed the spectral norm, eigenvalues and determinant of the matrix.

At this point, we can keep going to present the fundamental material which is about the definition of the generalized k-Horadam sequence { H k , n } n ∈ N. In fact, by [9], it was defined as the form

where, n ≥ 0, k ∈ R + and f 2 ( k ) + 4 g ( k ) > 0. Obviously, if we choose suitable values on f ( k ), g ( k ), a and b for (1), then this sequence reduces to the special all second-order sequences in the literature. For example, by taking f ( k ) = g ( k ) = 1, a = 0 and b = 1, the well-known Fibonacci sequence is obtained.

Binet’s formula allows us to express the generalized k-Horadam number in function of the roots α and β of the characteristic equation x 2 − f ( k ) x − g ( k ) = 0. Binet’s formula related to the sequence { H k , n } n ∈ N has the form

where X = b − a β and Y = b − a α.

Lemma 1 [9]

Let the entries of each matrix X n = ( H k , n − 1 H k , n H k , n H k , n + 1 ) be the generalized k-Horadam numbers. For n ≥ 1, we get

Definition 2 For any given c 0 , c 1 , c 2 , … , c n − 1 ∈ C, the r-circulant matrix C r = ( c i j ) n × n is defined by

Let C r = ( c 0 , c 1 , c 2 , … , c n − 1 ) denote an r-circulant matrix.

It is obvious that the matrix C r turns into a classical circulant matrix for r = 1. Let us take any A = [ a i j ] ∈ M n , n ( C ) (which could be a circulant matrix as well). The well-known Frobenius (or Euclidean) norm of the matrix A is given by

and also the spectral norm of A is presented by

where λ i ( A H A ) are the eigenvalues of A H A such that A H is the conjugate transpose of A. Then it is quite well known that

The following lemma will be needed in the proof of Theorem 5 below.

Lemma 3 [10]

For any matrices A = [ a i j ] ∈ M m , n ( C ) and B = [ b i j ] ∈ M m , n ( C ), we have

where A ∘ B is the Hadamard product and r 1 ( A ) = max 1 ≤ i ≤ n ∑ j = 1 n | a i j | 2, c 1 ( B ) = max 1 ≤ j ≤ n ∑ i = 1 n | b i j | 2.

Throughout this paper, the r-circulant matrix, whose entries are the generalized k-Horadam numbers, will be denoted by H = C r ( H k , 0 , H k , 1 , … , H k , n − 1 ).

In this paper, we first give lower and upper bounds for the spectral norms of H. In particular, by taking r = 1, we obtain lower and upper bounds for the spectral norms of the circulant matrix associated with the generalized k-Horadam numbers. Afterwards, we also formulate the eigenvalues and determinant of the matrix H defined above.

Lemma 4 For n ≥ 1, we have

where M = ( b − a f ( k ) ) 2 − a 2 − 2 X Y ( 1 − ( − g ( k ) ) n 1 + g ( k ) ), X = b − a β and Y = b − a α.

Proof Let A = ∑ i = 0 n − 1 H k , i 2. By using (1), we have

By considering (1), we obtain

which is desired. □

Theorem 5 Let H = C r ( H k , 0 , H k , 1 , H k , 2 , … , H k , n − 1 ) be an r-circulant matrix. Then we have

if | r | ≥ 1, then

if | r | < 1, then

where r ∈ C, ∑ i = 0 n − 1 H k , i 2 = H k , n 2 − g 2 ( k ) H k , n − 1 2 + M f 2 ( k ) + 2 g ( k ) − g 2 ( k ) − 1, M = ( b − a f ( k ) ) 2 − a 2 − 2 X Y ( 1 − ( − g ( k ) ) n 1 + g ( k ) ).

Proof The matrix H is of the form

and, by the definition of Frobenius norm, we clearly have

If | r | ≥ 1, then we obtain

Also, by considering Lemma 4, we can write

It follows that

where M = ( b − a f ( k ) ) 2 − a 2 − 2 X Y ( 1 − ( − g ( k ) ) n 1 + g ( k ) ). Then by (1), we have

Similarly, for | r | < 1, we can write

Again, by considering Lemma 4 and (1), we get

It follows that

Now, for | r | ≥ 1, we give the upper bound for the spectral norm of the matrix H as in the following. Let the matrices B and C be as

such that H = B ∘ C. Then we obtain

where M = ( b − a f ( k ) ) 2 − a 2 − 2 X Y ( 1 − ( − g ( k ) ) n 1 + g ( k ) ), X = b − a β and Y = b − a α. By considering Lemma 3, we can write

For | r | < 1, we give the upper bound for the spectral norm of the matrix H as in the following. On the other hand, if the matrices D and E are

such that H = D ∘ E, then we obtain

where M = ( b − a f ( k ) ) 2 − a 2 − 2 X Y ( 1 − ( − g ( k ) ) n 1 + g ( k ) ), X = b − a β and Y = b − a α. By Lemma 3, then we have

as required. □

Lemma 6 Let H be an n × n r-circulant matrix. Then we have

where w = e 2 π i n, i = − 1, j = 0 , 1 , … , n − 1.

Considering Lemma 6, we obtain the following theorem which gives us the eigenvalues of the matrix in (3).

Theorem 7 Let H = C r ( H k , 0 , H k , 1 , H k , 2 , … , H k , n − 1 ) be an r-circulant matrix. Then the eigenvalues of H are written by

where H k , n is the nth generalized k-Horadam number and w = e 2 π i n, i = − 1, j = 0 , 1 , … , n − 1.

Proof By Lemma 6, we have λ j ( H ) = ∑ i = 0 n − 1 H k , i r i n w − j i. Moreover, by (2), ∑ i = 0 n − 1 H k , i r i n w − j i = ∑ i = 0 n − 1 X α n − Y β n α − β r i n w − j i. Hence, we obtain

which completes the proof. □

Theorem 8 The determinant of H = C r ( H k , 0 , H k , 1 , H k , 2 , … , H k , n − 1 ) is formulated by

Proof It is clear that det ( H ) = ∏ j = 0 n − 1 λ j ( H ). By Theorem 7, we get

Also, by considering the well-known identity ∏ k = 0 n − 1 ( x − y w k ) = x n − y n, we can write

Thus the proof is completed, as desired. □

Remark 9 We should note that choosing suitable values on r, f ( k ), g ( k ), a and b in Theorems 5, 7 and 8, lower and upper bounds of the spectral norm, eigenvalues and determinant of r-circulant matrices for the special all second-order sequences are actually obtained.