ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.12813

J. Algebra Comb. Discrete Appl.
4(1) • 13–21

Received: 30 September 2015
Accepted: 10 April 2016

Journal of Algebra Combinatorics Discrete Structures and Applications

On the norms of r−circulant matrices with generalized
Fibonacci numbers

Research Article

Amara Chandoul

Abstract: In this paper, we obtain a generalization of [6, 8]. Firstly, we consider the so-called r−circulant ma-
trices with generalized Fibonacci numbers and then found lower and upper bounds for the Euclidean
and spectral norms of these matrices. Afterwards, we present some bounds for the spectral norms of
Hadamard and Kronecker product of these matrices.

2010 MSC: 15B05, 15A60, 65F35

Keywords: Matrix norm, r-circulant matrices, Generalized Fibonacci numbers

1. Introduction

The Fibonacci numbers are the elements of integer sequence {Fn}∞n=0 defined by the linear recurrence
equation 


F0 = 0, F1 = 1

Fn+2 = Fn+1 + Fn, for n = 0,1,2, ....

The sequence is s given by

0,1,1,2,3,5,8,13, . . . .

You cannot go very far in the lore of Fibonacci numbers without encountering the companion sequence of
Lucas numbers {Ln}∞n=0, which follows the same recursive pattern as the Fibonacci numbers, but begins
with other initial values. It is defined by the linear recurrence equation


L0 = 2, L1 = 1

Ln+2 = Ln+1 + Ln, for n = 0,1,2, ....

Amara Chandoul; Departamento de Matemática, Universidade de Brasília, Campus Universitário Darcy Ribeiro
Brasília - DF 70910-900, Brazil (email: amarachandoul@yahoo.fr).

13



A. Chandoul / J. Algebra Comb. Discrete Appl. 4(1) (2017) 13–21

The sequence is s given by

2,1,3,4,7,11,18,29, . . . .

The Fibonacci sequence has been studied extensively and generalized in many ways. Such generalizations
are possible in four directions, namely, by changing the initial values, by mixing two Lucas sequences, by
not demanding that the numbers in the sequences be integers, or by having more than two parameters.

One of generalization of Fibonacci i sequence is the integer sequence {Gn}∞n=0. It has the same
recurrence relation as Fibonacci and Lucas, namely it is defined by the linear recurrence equation


G0 = a, G1 = b

Gn+2 = Gn+1 + Gn, for n = 0,1,2, ....

The starting values of G0 = a and G1 = b can be specified. It therefore includes both sequences them
both as special cases.

This generalized Fibonacci sequence is giving by

a,b,(a + b),(a + 2b),(2a + 3b),(3a + 5b),(5a + 8b),(8a + 13b), ...

{Gn}∞n=0 is a generalization of {Fn}∞n=0 and {Ln}∞n=0. Furthermore, there is a relationship such that
Gn = aFn−1 + bFn, a,b ∈ R between Fibonacci and generalized Fibonacci numbers.

Fibonacci, Lucas numbers and their generalization have many intersting properties and applications
to almost every field of science and art. For the beauty and rich applications of these numbers and their
relatives one may see Vajda’s and Koshy’s books [7, 12].

Solak [10] defined the circulant matrices with Fibonacci and Lucas numbers, and he obtained lower
and upper bounds with the Fibonacci number for the Euclidean and spectral norms of these matrices.

Civciv and Türkmen [2] constructed the circulant matrix with the Lucas number and then presented
lower and upper bounds for the Euclidean and spectral norms of this matrix as a function of n and Ln
which is n th Lucas number. In [2], they are studied the norm bounds for the Hadamard inverse of this
matrix. In [1], authors computed some norms of r-circulant matrices associated with a number sequence.

In this study, we first construct the so-called r-circulant matrix with the generalized Fibonacci
numbers and then present some lower and upper bounds for the Euclidean and spectral norms of this
matrix.

We begin with some preliminaries related to our study. A matrix C = [cij] ∈ Mn(C) is called a
r-circulant matrix if it is of the form

cij =

{
cj−i j ≥ i
rcn+j−i j < i.

Obviously, the r-circulant matrix C is determined by parameter r and its first row elements
c0,c1, · · · ,cn−1. Especially, for r = 1, the matrix C is called a circulant matrix. For any A = [aij] ∈
Mm,n(C), the well known Frobenius (or Euclidean) norm of matrix A is

‖A‖E =

√√√√ n∑
i=1

n∑
j=1

|aij|2.

The spectral norm of the matrix A

‖A‖2 =
√

max
1≤i≤n

λi(A
HA)

14



A. Chandoul / J. Algebra Comb. Discrete Appl. 4(1) (2017) 13–21

where λi is an eigenvalue of AHA and AH is conjugate transpose of the matrix A. It is well known that

1
√
n
‖A‖E ≤‖A‖2 ≤‖A‖E.

If ‖.‖ is any norm on m×n complex matrices, then [5]

‖A◦B‖≤‖A‖‖B‖, (1)

where A◦B is the Hadamard product of A and B. Define the maximum column lenght norm c1(A) and
the maximum row lenght norm r1(A) of any matrix A respectively by

c1(A) = max
1≤j≤n

√√√√ n∑
i=1

|aij|2

and

r1(A) = max
1≤i≤n

√√√√ n∑
j=1

|aij|2.

Let A, B and C be m×n complex matrices. if A = B ◦C, then [4]

‖A‖2 ≤ r1(B)c1(C). (2)

2. Results

Definition 2.1. The r−circulant matrix A with generalized Fibonacci number is the matrix of the form

A =




G0 G1 . . . . . . Gn−1
rGn−1 G0 . . . . . . Gn−2

rGn−2 rGn−1
... . . . Gn−3

...
...

... G0
...

rG1 rG2 . . . rGn−1 G0


 . (3)

Theorem 2.2. Let A be the matrix defined in (3). Then

(i) If |r| ≥ 1, then

‖A‖E ≥
√
n(Gn−1Gn −G1G0 + G20), ‖A‖2 ≥

√
Gn−1Gn −G1G0 + G20

and

‖A‖2 ≤ min




√
((n−1)|r|2 + 1)(Gn−1Gn −G1G0 + G20),√
((n−1)|r|2 + G20)(Gn−1Gn −G1G0 + 1),√
(Gn−1Gn −G1G0 + G20)(Gn−1Gn −G1G0 + 1).

Moreover, if G0 = 0, then √
GnGn−1 ≤‖A‖2 ≤ |r|Gn−1Gn.

15



A. Chandoul / J. Algebra Comb. Discrete Appl. 4(1) (2017) 13–21

(ii) If |r| < 1, then

‖A‖E ≥ |r|
√
n(Gn−1Gn −G1G0 + G20), ‖A‖2 ≥ |r|

√
Gn−1Gn −G1G0 + G20

and

‖A‖2 ≤ min




√
n(Gn−1Gn −G1G0 + G20),√
((n−1) + G20)(Gn−1Gn −G1G0 + 1),√
((n−1) + G20)(Gn−1Gn −G1G0 + G20).

Moreover, if G0 = 0, then

|r|
√
GnGn−1 ≤‖A‖2 ≤

√
(n−1)Gn−1Gn.

Proof. From (3), we have

‖A‖E =

√√√√n−1∑
k=0

(n−k)G2k +
n−1∑
k=1

k|r|2G2k,

Let matrices be defined as

B1 = (b1,ij) =

{
b1,ij = r, i > j
b1,ij = 1, i ≤ j

B2 = (b2,ij) =

{
b2,ij = G(mod(j−i,n)), i ≥ j
b2,ij = 1, i < j

B3 = (b3,ij) =



b3,ij = r, i > j
b3,ij = G0, i = j
b3,ij = 1, i < j

C1 = (c1,ij) = (G(mod(j−i,n))).

C2 = (c2,ij) =



c2,ij = r, i > j
c2,ij = 1, i = j
c2,ij = G(mod(j−i,n)), i < j

and

C3 = (c3,ij) =

{
c3,ij = G(mod(j−i,n)), i 6= j
c3,ij = 1, i = j

such that A = Bk ◦Ck, 1 ≤ k ≤ 3. Then we have

r1(B1) = max
1≤i≤n

√√√√ n∑
j=1

|b1,ij|2 =



√
(n−1)|r|2 + 1, |r| ≥ 1
√
n, |r| < 1.

r1(B2) = max
1≤i≤n

√√√√ n∑
j=1

|b2,ij|2 =
√
Gn−1Gn −G1G0 + G20.

r1(B3) = max
1≤i≤n

√√√√ n∑
j=1

|b3,ij|2 =



√
G20 + (n−1)|r|2, |r| ≥ 1√
G20 + (n−1), |r| < 1.

(4)

16



A. Chandoul / J. Algebra Comb. Discrete Appl. 4(1) (2017) 13–21

c1(C1) = max
1≤j≤n

√√√√ n∑
i=1

|c1,ij|2 =
√
Gn−1Gn −G1G0 + G20.

c1(C2) = max
1≤j≤n

√√√√ n∑
i=1

|c2,ij|2 =
√
Gn−1Gn −G1G0 + 1.

c1(C3) = max
1≤j≤n

√√√√ n∑
i=1

|c3,ij|2 =
√
Gn−1Gn −G1G0 + 1.

(5)

If G0 = 0, we consider the matrices

B4 = (b4,ij) =

{
b4,ij = rG(mod(j−i,n)), i ≥ j
b4,ij = 1, i < j.

C4 = (c4,ij) =

{
c4,ij = G(mod(j−i,n)), i ≤ j
c4,ij = 1, i > j.

Then, we have

r1(B4) = max
1≤i≤n

√√√√ n∑
j=1

|b4,ij|2 = |r|
√
Gn−1Gn,

c1(C4) = max
1≤j≤n

√√√√ n∑
i=1

|c4,ij|2 =
√
Gn−1Gn.

(6)

(i) If |r| ≥ 1, then we have

‖A‖E ≥
√
n(Gn−1Gn −G1G0 + G20)

Thus, we obtain from
1
√
n
‖A‖E ≤‖A‖2,

‖A‖2 ≥
√
Gn−1Gn −G1G0 + G20

On the other hand, using (4), (5) and (2), we have

‖A‖2 ≤ min{r1(B1)c1(C1),r1(B2)c1(C2),r1(B3)c1(C3)}.

Thus, we have

‖A‖2 ≤ min




√
((n−1)|r|2 + 1)(Gn−1Gn −G1G0 + G20),√
((n−1)|r|2 + G20)(Gn−1Gn −G1G0 + 1),√
(Gn−1Gn −G1G0 + G20)(Gn−1Gn −G1G0 + 1).

Moreover, if G0 = 0, using (6), then√
GnGn−1 ≤‖A‖2 ≤ |r|Gn−1Gn.

(ii) If |r| < 1, then we have

‖A‖E ≥
√
n|r|2(Gn−1Gn −G1G0 + G20).

17



A. Chandoul / J. Algebra Comb. Discrete Appl. 4(1) (2017) 13–21

Thus, we have from
1
√
n
‖A‖E ≤‖A‖2,

‖A‖2 ≥ |r|
√

(Gn−1Gn −G1G0 + G20).

On the other hand, using (4), (5) and (2), we have

‖A‖2 ≤ min{r1(B1)c1(C1),r1(B2)c1(C2),r1(B3)c1(C3)}.

Thus, we have

‖A‖2 ≤ min




√
n(Gn−1Gn −G1G0 + G20),√
((n−1) + G20)(Gn−1Gn −G1G0 + 1),√
((n−1) + G20)(Gn−1Gn −G1G0 + G20).

Moreover, if G0 = 0, then

|r|
√
GnGn−1 ≤‖A‖2 ≤

√
(n−1)Gn−1Gn.

Corollary 2.3. Let G0 = a = 0 and G1 = b = 1 be in the theorem. Then

(i) If |r| ≥ 1, then
√
FnFn−1 ≤‖A‖2 ≤ |r|FnFn−1.

(ii) If |r| < 1, then |r|
√
FnFn−1 ≤‖A‖2 ≤

√
(n−1)FnFn−1.

Corollary 2.4. Let G0 = a = 2 and G1 = b = 1 be in the theorem. Then

(i) If |r| ≥ 1, then

√
LnLn−1 + 2 ≤‖A‖2 ≤ min




√
((n−1)|r|2 + 1)(LnLn−1 + 2),√
((n−1)|r|2 + 4)(LnLn−1 −1),√
(LnLn−1 + 2)(LnLn−1 −1).

(ii) If |r| < 1, then

|r|
√
LnLn−1 + 2 ≤‖A‖2 ≤

√
(n + 3)(LnLn−1 −1)

Theorem 2.5. Let A be the matrix defined in (3). Then

(i) If |r| ≥ 1, then

‖A‖2 ≥



√
a2(1 + Fn−2Fn−1) + 2abF

2
n−1 + b

2FnFn−1, n odd

√
a2(1 + Fn−2Fn−1) + 2ab(F

2
n−1 −1) + b2FnFn−1, n even

(ii) If |r| < 1, then

‖A‖2 ≥



|r|
√
a2(1 + Fn−2Fn−1) + 2abF

2
n−1 + b

2FnFn−1, n odd

|r|
√
a2(1 + Fn−2Fn−1) + 2ab(F

2
n−1 −1) + b2FnFn−1, n even

18



A. Chandoul / J. Algebra Comb. Discrete Appl. 4(1) (2017) 13–21

Proof. We have,

n−1∑
k=0

G2k = Gn−1Gn −G1G0 + G20

=

n−1∑
k=0

(aFk−1 + bFk)
2 = a2

n−1∑
k=0

F2k−1 + 2ab

n−1∑
k=0

Fk−1Fk + b
2
n−1∑
k=0

F2k .

As

n−1∑
k=0

F2k−1 = Fn−1Fn−2 + 1,

n−1∑
k=0

F2k = FnFn−1

and

n−1∑
k=0

Fk−1Fk =



F2n−1, n odd

F2n−1 −1, n even

The proof of Theorem 2.5 become trivial.

Theorem 2.6. Let A be the matrix defined in (3). Then

(i) If |r| ≥ 1, then

‖A‖2 ≤




min




√
a2(1 + Fn−2Fn−1) + 2abF

2
n−1 + b

2FnFn−1.√
(n−1)|r|2 + 1,√
a2Fn−2Fn−1 + 2abF

2
n−1 + b

2FnFn−1 + 1.√
(n−1)|r|2 + a2,√
a2(1 + Fn−2Fn−1) + 2abF

2
n−1 + b

2FnFn−1.√
a2Fn−2Fn−1 + 2abF

2
n−1 + b

2FnFn−1 + 1.

n odd

min




√
a2(1 + Fn−2Fn−1) + 2ab(F

2
n−1 −1) + b2FnFn−1.√

(n−1)|r|2 + 1,√
a2Fn−2Fn−1 + 2ab(F

2
n−1 −1) + b2FnFn−1 + 1.√

(n−1)|r|2 + a2,√
a2(1 + Fn−2Fn−1) + 2ab(F

2
n−1 −1) + b2FnFn−1.√

a2Fn−2Fn−1 + 2ab(F
2
n−1 −1) + b2FnFn−1 + 1.

n even

19



A. Chandoul / J. Algebra Comb. Discrete Appl. 4(1) (2017) 13–21

(ii) If |r| < 1, then

‖A‖2 ≤




min




√
n(a2(1 + Fn−2Fn−1) + 2abF

2
n−1 + b

2FnFn−1),

√
a2Fn−2Fn−1 + 2abF

2
n−1 + b

2FnFn−1 + 1.√
(n−1) + a2,√
a2(1 + Fn−2Fn−1) + 2abF

2
n−1 + b

2FnFn−1.√
(n−1) + a2,

n odd

min




√
n(a2(1 + Fn−2Fn−1) + 2ab(F

2
n−1 −1) + b2FnFn−1),√

a2Fn−2Fn−1 + 2ab(F
2
n−1 −1) + b2FnFn−1 + 1.√

(n−1) + a2,√
a2(1 + Fn−2Fn−1) + 2ab(F

2
n−1 −1) + b2FnFn−1.√

(n−1) + a2.

n even

Proof. As

n−1∑
k=1

F2k−1 = Fn−1Fn−2,

n−1∑
k=1

F2k = FnFn−1

and

n−1∑
k=1

Fk−1Fk =



F2n−1, n odd

F2n−1 −1, n even

The proof of Theorem 2.6 become trivial.

Corollary 2.7. Let G0 = a = 2 and G1 = b = 1 be in the theorem. As FnFn−1 ≥ n−1, ∀n ≥ 0 and as
we have FnFn−1 = F2n−1 + Fn−2Fn−1, then, we obtain

(i) If |r| ≥ 1, then

√
5FnFn−1 + 4 ≤‖A‖2 ≤

√
((n−1)|r|2 + 4)(5FnFn−1 + 1), n odd√

5FnFn−1 ≤‖A‖2 ≤
√
((n−1)|r|2 + 4)(5FnFn−1 −3), n even

(ii) If |r| < 1, then

|r|
√
5FnFn−1 + 4 ≤‖A‖2 ≤

√
n(5FnFn−1 + 4), n odd

|r|
√
5FnFn−1 ≤‖A‖2 ≤

√
n(5FnFn−1), n even

Acknowledgment: We would like to thank Saäd Chandoul and Massöuda Loörayed for helpful
discussions and many remarks.

20



A. Chandoul / J. Algebra Comb. Discrete Appl. 4(1) (2017) 13–21

References

[1] D. Bozkurt, T. Y. Tam, Determinants and inverses of r−circulant matrices associated with a number
sequences, Linear and Multilinear Algebra 63(10) (2015) 2079–2088.

[2] H. Civciv, R. Turkmen, Notes on norms of circulant matrices with Lucas numbers, Int. J. Inf. Syst.
Sci. 4(1) (2008) 142–147.

[3] C. He, J. Ma, K. Zhang, Z. Wang, The upper bound estimation on the spectral norm of r−circulant
matrices with the Fibonacci and Lucas numbers, J. Inequal. Appl. Article ID 72 (2015) 1–10.

[4] R. A. Horn, C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge,
1991.

[5] R. Mathias, The spectral norm of nonnegative matrix, Linear Algebra Appl. 139 (1990), 269–284.
[6] A. Nalli, M. Sen, On the norms of circulant matrices with generalized Fibonacci numbers, Selçuk J.

Appl. Math. 11(1) (2010) 107–116.
[7] T. Koshy, Fibonacci and Lucas Numbers with Application, John Wiley and Sons, Inc., 2001.
[8] S. Shen, J. Cen, On the norms of r−circulant matrices with Fibonacci and Lucas numbers, Appl.

Math. Comput. 216(10) (2010) 2891–2897.
[9] S. Solak, On the norms of circulant matrices with Fibonacci and Lucas numbers, Appl. Math.

Comput. 160(1) (2005) 125–132.
[10] S. Solak, Erratum to "On the norms of circulant matrices with Fibonacci and Lucas numbers" [Appl.

Math. Comput. 160(1) (2005) 125–132], Appl. Math. Comput. 190(2) (2007) 1855–1856.
[11] N. Tuglu, C. Kizilates, On the norms of circulant and r−circulant matrices with the hyperharmonic

Fibonacci numbers, J. Inequal. Appl. Article ID 253 (2015) 1–11.
[12] S. Vajda, Fibonacci and Lucas numbers and Golden Section, John Wiley and Sons, Inc., 1989.

21

http://dx.doi.org/10.1080/03081087.2014.941291
http://dx.doi.org/10.1080/03081087.2014.941291
http://www.ams.org/mathscinet-getitem?mr=2401770
http://www.ams.org/mathscinet-getitem?mr=2401770
http://dx.doi.org/10.1186/s13660-015-0596-5
http://dx.doi.org/10.1186/s13660-015-0596-5
http://dx.doi.org/10.1016/0024-3795(90)90403-Y
http://www.ams.org/mathscinet-getitem?mr=2674613
http://www.ams.org/mathscinet-getitem?mr=2674613
http://dx.doi.org/10.1016/j.amc.2010.03.140
http://dx.doi.org/10.1016/j.amc.2010.03.140
http://dx.doi.org/10.1016/j.amc.2003.08.126
http://dx.doi.org/10.1016/j.amc.2003.08.126
http://dx.doi.org/10.1016/j.amc.2007.02.075
http://dx.doi.org/10.1016/j.amc.2007.02.075
http://dx.doi.org/10.1186/s13660-015-0778-1
http://dx.doi.org/10.1186/s13660-015-0778-1

	Introduction
	Results
	References