# The Identical Estimates of Spectral Norms for Circulant Matrices with Binomial Coefficients Combined with Fibonacci Numbers and Lucas Numbers Entries

Authors: Jianwei Zhou
Journal: Journal of Function Spaces
 (2014)

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## Abstract

Improved estimates for spectral norms of circulant matrices are investigated, and the entries are binomial coefficients combined with either Fibonacci numbers or Lucas numbers. Employing the properties of given circulant matrices, this paper improves the inequalities for their spectral norms, and gets corresponding identities of spectral norms. Moreover, by some well-known identities, the explicit identities for spectral norms are obtained. Some numerical tests are listed to verify the results.


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## Body

## 1. Introduction

Circulant matrices have connection to physics, signal and image processing, probability, statistics, numerical analysis, algebraic coding theory, and many other areas. There are lots of examples from statistical signal processing and information theory that illustrate the application of the circulant matrices, which emphasize how the asymptotic eigenvalue distribution theorem allows one to evaluate results for processes (for the details please refer to [1–3] and the reference therein). Meanwhile a real circulant stochastic process can be described with autocovariance matrices, which are subjected to a cyclical permutation. With the help of autocovariance circulant matrices, it is easy to provide derivations of some results that are central to the analysis of statistical periodograms and empirical spectral density functions (see [4]).In past decades, the estimates for spectral norms of matrices have been investigated in lots of literatures. Moreover, the determinants and inverses of circulant matrices are stated in many articles. The norms of circulant matrices play an important role in analysing the process of statistics, numerical analysis, and many other problems (for more details, please refer to [3, 5–10] and the reference therein). Bryc and Sethuraman [11] investigated the maximum eigenvalue for circulant matrices. Solak [7] obtained lower and upper bounds for the spectral norm of circulant matrices, where the entries are classical Fibonacci numbers. İpek [8] establishes spectral norms of circulant matrices with Fibonacci and Lucas numbers. Furthermore, circulant matrices take up an important status in stochastic calculus, Meckes [12, 13] gave some results on the spectral norm of a special random Toeplitz matrix and random circulant matrices, Mehta [14] made a deep discussion on random circulant matrices.The outline of this paper is as follows. In Section2, we state some preliminaries and recall some well-known results. In Section 3, we focus on the identities of estimations for spectral norms. In Section 4, we present various numerical examples to exhibit the accuracy and efficiency of our results. Finally, we summarise this paper and illustrate our future work.


## 2. Preliminaries

The Fibonacci and Lucas sequences{
F
n
} and {
L
n
} are defined by the recurrence relations:
(1)
F
n
=
F
n
-
1
+
F
n
-
2
,
L
n
=
L
n
-
1
+
L
n
-
2
,
with F
0
=
0, F
1
=
1, L
0
=
2, and L
1
=
1, respectively.Obviously, the Fibonacci and Lucas sequences are listed in the following sequence:(2)
n
0
1
2
3
4
5
6
7
⋯
F
n
0
1
1
2
3
5
8
13
⋯
L
n
2
1
3
4
7
11
18
29
⋯
,
and their corresponding Binet forms are (see [15])
(3)
F
n
=
1
5
[
(
1
+
5
2
)
n
-
cos
⁡
⁡
(
π
n
)
(
1
+
5
2
)
-
n
]
,
L
n
=
(
1
+
5
2
)
n
+
cos
⁡
⁡
(
π
n
)
(
1
+
5
2
)
-
n
.Now, we recall that, for(4)
B
=
(
F
0
F
1
⋯
F
n
-
1
F
n
-
1
F
0
⋯
F
n
-
2
F
n
-
2
F
n
-
1
⋯
F
n
-
3
⋮
⋮
⋱
⋮
F
1
F
2
⋯
F
0
)
n
×
n
,
there hold the following estimates:
(5)
F
n
F
n
-
1
+
4
(
F
n
-
1
2
+
F
n
-
1
F
n
-
2
+
1
-
(
-
1
)
n
2
)
≤
∥
B
∥
2
,
∥
B
∥
2
≤
F
n
F
n
-
1
+
4
(
F
n
-
1
2
+
F
n
-
1
F
n
-
2
+
1
-
(
-
1
)
n
2
)
×
1
+
F
n
F
n
-
1
+
4
(
F
n
-
1
2
+
F
n
-
1
F
n
-
2
-
3
+
(
-
1
)
n
2
)
.
For the details please refer to [7].There are lots of identities for Fibonacci numbers and Lucas numbers combined with Binomial coefficients (for more details please refer to [8, 16–18] and the reference therein). In this paper, we focus on the following identities:
(6)
∑
i
=
0
n
(
n
i
)
F
i
=
F
2
n
,
∑
i
=
0
n
(
n
i
)
L
i
=
L
2
n
,
∑
i
=
0
n
(
n
i
)
F
i
+
k
=
F
2
n
+
k
,
∑
i
=
0
n
(
n
i
)
L
i
+
k
=
L
2
n
+
k
.
Furthermore, for all k
∈
ℤ
+, there hold the following identities:
(7)
∑
i
=
0
n
(
n
i
)
F
2
i
+
k
=
{
5
n
/
2
F
n
+
k
,
(
n
even
)
,
5
(
n
-
1
)
/
2
L
n
+
k
,
(
n
odd
)
,
(8)
∑
i
=
0
n
(
n
i
)
L
2
i
+
k
=
{
5
n
/
2
L
n
+
k
(
n
even
)
,
5
(
n
+
1
)
/
2
F
n
+
k
(
n
odd
)
.Definition 1 (see [19]).
A circulant matrix is ann
×
n complex matrix with the following form:
(9)
A
=
(
a
0
a
1
⋯
a
n
-
1
a
n
-
1
a
0
⋯
a
n
-
2
a
n
-
2
a
n
-
1
⋯
a
n
-
3
⋮
⋮
⋱
⋮
a
1
a
2
⋯
a
0
)
n
×
n
.
The first row ofA is (
a
0
,
a
1
,
…
,
a
n
-
1
) and its (
j
+
1
)th row is obtained by giving its jth row a right circular shift by one positions.Definition 2 (see [3]).
The spectral norm∥
·
∥
2 of a matrix A with complex entries is the square root of the largest eigenvalue of the positive semidefinite matrix A
*
A:
(10)
∥
A
∥
2
=
λ
max
⁡
(
A
*
A
)
,
where A
* denotes the conjugate transpose of A. Therefore if A is an n
×
n real symmetric matrix or A is a normal matrix, then
(11)
∥
A
∥
2
=
max
⁡
1
≤
i
≤
n
⁡
|
λ
i
|
,
where λ
1
,
λ
2
,
…
,
λ
n are the eigenvalues of A.


## 3. The Identities of Estimations for Spectral Norms

We give the main theorems of this paper in the following parts.Theorem 3.
LetB
1 be as the matrix in (9), and let the first row of B
1 be (
(
n
0
)
F
0
,
(
n
1
)
F
1
,
…
,
(
n
n
)
F
n
). Then one has
(12)
∥
B
1
∥
2
=
F
2
n
.Proof.
Combining with Definition2, the spectral radius of B
1 is equal to its spectral norm, where we used the fact that B
1 is normal. Moreover, by the irreducible and entrywise nonnegative properties, we deduce that ∥
B
1
∥
2 is equal to its Perron value. Denote by v
=
(
1,1
,
…
,
1
)
T an n-dimensional column vector. There holds
(13)
B
1
v
=
(
∑
i
=
0
n
(
n
i
)
F
i
)
v
.
Obviously,∑
i
=
0
n
‍
(
n
i
)
F
i is an eigenvalue of B
1 associated with the positive eigenvector v, which is the Perron value of B
1. Employing the first identity in (6), we have
(14)
∥
B
1
∥
2
=
F
2
n
.
This completes the proof.With the same approach, we obtain the following corollary.Corollary 4.
LetB
2 be as the matrix in (9), and let the first row of B
2 be (
(
n
0
)
L
0
,
(
n
1
)
L
1
,
…
,
(
n
n
)
L
n
). Then one has the following identity:
(15)
∥
B
2
∥
2
=
L
2
n
.Theorem 5.
LetB
3 be with the form as (9). For all k
∈
ℤ
+, if the first row of B
3 is (
(
n
0
)
F
k
,
(
n
1
)
F
1
+
k
,
…
,
(
n
n
)
F
n
+
k
), then one obtains
(16)
∥
B
3
∥
2
=
F
2
n
+
k
.Proof.
Following the same techniques of the above theorem and combining with the fact thatB
3 is irreducible and entrywise nonnegative, we declare that the spectral norm of B
3 is equal to its Perron value. Let v
T
=
(
1,1
,
…
,
1
)
1
×
n. Then
(17)
B
3
v
=
(
∑
i
=
0
n
(
n
i
)
F
i
+
k
)
v
.
Obviously, we declare that∑
i
=
0
n
‍
(
n
i
)
F
i
+
k is an eigenvalue of B
3 associated with v. With simple analysis, we obtain that ∑
i
=
0
n
‍
(
n
i
)
F
i
+
k is equal to the Perron value of B
3. Combining with the third identity of binomial coefficients and Fibonacci numbers in (6), we obtain
(18)
∥
B
3
∥
2
=
F
2
n
+
k
,
which completes the proof.Similarly, there holds the following corollary.Corollary 6.
LetB
4 be as the matrix in (9). For all k
∈
ℤ
+, the first row of B
4 is (
(
n
0
)
L
k
,
(
n
1
)
L
1
+
k
,
…
,
(
n
n
)
L
n
+
k
); then
(19)
∥
B
4
∥
2
=
L
2
n
+
k
.Now, we are at the point to recall the following lemma to verify the identities of spectral norms with other approaches.Lemma 7 (see [3]).
LetA be a nonnegative matrix. If the column sums of A are equal, then
(20)
ρ
(
A
)
=
∥
A
∥
1
,
where ρ
(
A
)
=
max
⁡
⁡
{
|
λ
|
:
λ
i
s
a
n
e
i
g
e
n
v
a
l
u
e
o
f
m
a
t
r
i
x
A
} and ∥
·
∥
1 denotes the maximum column sum matrix norm.Theorem 8.
LetB
5 be with the form as (9) and let the first row of B
5 be (
(
n
0
)
F
k
,
(
n
1
)
F
2
+
k
,
…
,
(
n
n
)
F
2
n
+
k
). Then one deduces the following identity:
(21)
∥
B
5
∥
2
=
{
5
n
/
2
F
n
+
k
(
n
e
v
e
n
)
,
5
(
n
-
1
)
/
2
L
n
+
k
(
n
o
d
d
)
,
where k
∈
ℤ
+.Proof.
Obviously, the circulant matrixB
5 is normal; with the results of Definition 2, we declare that the spectral radius of B
5 is equal to ρ
(
B
5
); that is, ∥
B
5
∥
2
=
ρ
(
B
5
). Furthermore, applying entrywise nonnegative properties and column sum of B
5 are certain constant K
col, which is described in (7). By Lemma 7, we obtain
(22)
ρ
(
B
5
)
=
K
col
.
Employing the identities of Fibonacci numbers and Binomial coefficients in (7), we have
(23)
∥
B
5
∥
2
=
{
5
n
/
2
F
n
+
k
(
n
even
)
,
5
(
n
-
1
)
/
2
L
n
+
k
(
n
odd
)
.
This completes the proof.Furthermore, we give the following corollary without proofs, which can be proved with the same approaches as the above theorem.Corollary 9.
LetB
6 be as the matrix in (9). For all k
∈
ℤ
+, the first row of B
6 is (
(
n
0
)
L
k
,
(
n
1
)
L
2
+
k
,
…
,
(
n
n
)
L
2
n
+
k
); then we have the following identity:
(24)
∥
B
6
∥
2
=
{
5
n
/
2
L
n
+
k
(
n
e
v
e
n
)
,
5
(
n
+
1
)
/
2
F
n
+
k
(
n
o
d
d
)
.


## 4. Numerical Examples

In this section, we give some examples to verify our identities in the above theorems and corollaries.Example 1.
In this example, we give the numerical results forB
i
(
i
=
1,2
,
3,4
) in Table 1.Table 1
Spectral norms ofB
i (i
=
1,2
,
3,4) and k
=
1.
n
1
2
3
4
5
6
7
8
∥
B
1
∥
2
1
3
8
21
55
144
377
987
F
2
n
1
3
8
21
55
144
377
987
∥
B
2
∥
2
3
7
18
47
123
322
843
2207
L
2
n
3
7
18
47
123
322
843
2207
∥
B
3
∥
2
2
5
13
34
89
233
610
1597
F
2
n
+
1
2
5
13
34
89
233
610
1597
∥
B
4
∥
2
4
11
29
76
199
521
1364
3571
L
2
n
+
1
4
11
29
76
199
521
1364
3571Example 2.
For simplicity, letk
=
1. We give the numerical results for B
5 and B
6 in Table 2.
With the data in Tables1 and 2, we declare that the identity for the spectral norm of B
i
(
i
=
1
,
…
,
6
) holds.Table 2
Spectral norms ofB
6 and B
6.
n
1
2
3
4
5
6
7
8
∥
B
5
∥
2
3
10
35
125
450
1625
5875
21250
5
n
/
2
F
n
+
1
∖
10
∖
125
∖
1625
∖
21250
5
(
n
-
1
)
/
2
L
n
+
1
3
∖
35
∖
450
∖
5875
∖
∥
B
6
∥
2
5
20
100
275
1000
3625
13125
47500
5
(
n
+
1
)
/
2
F
n
+
1
5
∖
100
∖
1000
∖
13125
∖
5
n
/
2
L
n
+
1
∖
20
∖
275
∖
3625
∖
47500


## 5. Conclusion

This paper had discussed the identical estimates of spectral norms for some circulant matrices, which are listed by explicit formulations. In the future, we are going to investigate the determinants, inverses of circulant matrices with certain entries, and, inspired by [6], we will investigate the properties of g-circulant matrices. Particularly worth mentioning is the fact that, for the g-circulant matrix, we had some numerical results to prove the fact that the same identical estimates hold precisely, and we will concern on the theoretical confirmation in part of the future work.



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