Bounds For The Norms Of Toeplitz Matrices With K-Jacobsthal And k-Jacobsthal Lucas Numbers

RESEARCH ARTICLE

90

BOUNDS FOR THE NORMS OF TOEPLITZ MATRICES WITH JACOBSTHAL AND k-JACOBSTHAL LUCAS NUMBERS

Şükran UYGUN1*_{, Hülya AYTAR}2

1_{Gaziantep University, Science and Arts Faculty, Departments of Mathematics, Gaziantep, suygun@gantep.edu.tr,} ORCID: 0000-0002-7878-2175

2 _{Gaziantep University, Science and Arts Faculty, Departments of Mathematics, Gaziantep, aytarhulya93@gmail.com,} ORCID: 0000-0002-1430-1782

Received Date: 12.04.2020 Accepted Date:04.09.2020

ABSTRACT

This work is concerned with the spectral, Euclid norms of Toeplitz matrices with generalized 𝑘- Jacobsthal and k- Jacobsthal Lucas entries. 𝑘- Jacobsthal and k- Jacobsthal Lucas sequences are two generalizations of two very popular special integer sequences called Jacobsthal and Jacobsthal Lucas sequences. Upper and lower bounds for the spectral norms of these matrices, that is, the matrices of the forms 𝐴 = 𝑇 (𝑗_𝑘,0 , 𝑗_𝑘,1 , … , 𝑗_{𝑘,𝑛−1} ) and 𝐵 = 𝑇 (𝑐_𝑘,0 , 𝑐_𝑘,1 , … , 𝑐_{𝑘,𝑛−1} ) are obtained. The upper bounds for the Euclidean and spectral norms of Kronecker and Hadamard product matrices of Toeplitz matrices with k-Jacobsthal and the k- Jacobsthal Lucas numbers are computed.

Keywords: Hadamard product, k -Jacobsthal numbers, k -Jacobsthal Lucas numbers, Kronecker

product, Norm, Toeplitz matrix

1. INTRODUCTION

Special matrices is a widely studied subject in matrix analysis. Especially special matrices whose entries are well-known number sequences have become a very interesting research subject in recent years and many authors have obtained some good results in this area. Sequences are the building blocks of special matrices such as circulant, Toeplitz, Hankel, geometric matrices. There have been many papers about the norms of special matrices. Recently, there has been much interest in investigation of some special matrices. Because of this, various number sequences are used as entries, and the properties of the resulting matrices are investigated. Research on these special matrices normally would revolve around the investigation of their determinants, eigenvalues, norms, inverses and bounds of norms.

In [6], the authors have studied bounds of the spectral norms of circulant matrices with Fibonacci numbers. In [7], Akbulak and Bozkurt studied the norms of Toeplitz matrices involving Fibonacci and Lucas numbers. Shen [8] investigated the upper and lower bounds for the spectral norms of Toeplitz matrices involving k-Fibonacci and k-Lucas numbers. In [9], Daşdemir demonsrated the norms of Toeplitz matrices with the Pell, Pell-Lucas and modified Pell numbers. Kocer [10] has given some properties of the modified Pell, Jacobsthal and Jacobsthal-Lucas numbers, then she has defined the

Uygun, Ş. and Aytar, H., Journal of Scientific Reports-A, Number 45, 90-100, December 2020.

91

circulant, negacyclic and semicirculant matrices with these numbers and she has investigated the norms, eigenvalues and determinants of these matrices. Raza and Ali [11] studied on the norms of some special matrices with generalized Fibonacci sequence. Uygun, constructed bounds for the norms of circulant matrices with the k -Jacobsthal and k -Jacobsthal Lucas numbers [13].

Considering the above articles, on the one hand, we obtain new lower and upper bounds estimates for the spectral norms of Toeplitz matrices with k-Jacobsthal and k-Jacobsthal Lucas numbers. Furthermore, Euclidean norms, and maximum row and maximum column norms of Toeplitz matrices with k-Jacobsthal and k-Jacobsthal Lucas numbers are computed. Then bounds for Euclidean and sprectral norms of Kronecker and Hadamard product of the matrices are calculated.

Now, we give some fundamental information related to our study. For n∈Z, the classic Jacobsthal and Jacobsthal Lucas sequences are defined respectively by the second order homogeneous linear recurrence relations

𝒋𝒏+𝟐= 𝒋𝒏+𝟏+ 𝟐𝒋𝒏 , 𝒋𝟎= 𝟎 , 𝒋𝟏= 𝟏,

𝒄𝒏+𝟐= 𝒄𝒏+𝟏+ 𝟐𝒄𝒏 , 𝒄𝟎= 𝟐 , 𝒄𝟏= 𝟏.

Many generalizations of the well-known Jacobsthal sequence have been introduced and studied. For example, the generalized k -Jacobsthal and k -Jacobsthal Lucas numbers have been studied [12]. For 𝒏 ≥ 𝟐, 𝒏 ∈ ℕ , the k-Jacobsthal {𝒋𝒌,𝒏}_𝒏∈ℕ, and the k-Jacobsthal Lucas {𝒄𝒌,𝒏}_𝒏∈ℕ sequences are defined

recurrently by

𝒋𝒌,𝒏= 𝒌 𝒋𝒌,𝒏−𝟏+ 𝟐 𝒋𝒌,𝒏−𝟐 , 𝒋𝒌,𝟎= 𝟎 , 𝒋𝒌,𝟏= 𝟏 , (1)

𝑐𝑘,𝑛= 𝑘 𝑐𝑘,𝑛−1+ 2 𝑐𝑘,𝑛−2, 𝑐𝑘,0= 2 , 𝑐𝑘,1= 𝑘. (2)

respectively in [12]. The first k-Jacobsthal numbers are 0, 1, k, 𝒌𝟐+ 𝟐, 𝒌𝟑+ 𝟒𝒌, 𝒌𝟒+ 𝟔𝒌𝟐+ 𝟒 …The first k- Jacobsthal Lucas numbers are 2 , k , 𝒌𝟐_{+ 𝟒 , 𝒌}𝟑_{+ 𝟔𝒌 , 𝒌}𝟒_{+ 𝟖𝒌}𝟐_{+ 𝟖…}

Recurrences (1) ve (2) involve the characteristic equation 𝑥2− 𝑘𝑥 − 2 = 0 with roots 𝛼 =

𝑘+ √𝑘2₊₈

2 , 𝛽 =

𝑘− √𝑘2₊₈

2 . Binet’s formulas of the k-Jacobsthal and k- Jacobsthal Lucas are defined

respectively by 𝑗𝑘,𝑛= 𝛼

𝑛_−𝛽𝑛

𝛼−𝛽 , 𝑐𝑘,𝑛= 𝛼

𝑛_{+ 𝛽}𝑛_{. (3)}

Extension to negative values of n can be made, k-Jacobsthal and k-Jacobsthal Lucas sequence with negative indices are demonstrated by

𝑗𝑘,−𝑖 =(−1) 𝑖+1_𝑗 𝑘,𝑖 2𝑖 , 𝑐𝑘,−𝑖= (−1)𝑖𝑐_𝑘,𝑖 2𝑖 .

An 𝑛 × 𝑛 matrix 𝑇 = {𝑡𝑖𝑗} ∈ 𝑀𝑛(𝐶) is called a Toeplitz matrix if it is of the form 𝑡𝑖𝑗= 𝑡𝑖−𝑗 for

Uygun, Ş. and Aytar, H., Journal of Scientific Reports-A, Number 45, 90-100, December 2020. 92 𝑇𝑛= [ 𝑡0 𝑡−1 𝑡−2 ⋯ 𝑡1−𝑛 𝑡1 𝑡0 𝑡−1 ⋯ 𝑡2−𝑛 𝑡2 𝑡1 𝑡0 ⋯ 𝑡3−𝑛 ⋮ ⋮ ⋮ ⋱ ⋮ 𝑡𝑛−1 𝑡𝑛−2 𝑡𝑛−3 ⋯ 𝑡0 ] (4)

Toeplitz matrix is determined by its first row (or column).

For any 𝐴 = [𝑎𝑖𝑗]𝜖𝑀{𝑚,𝑛}(𝐶), the largest absolute column sum (1-norm) and the largest absolute row

sum(∞-norm) norms are ‖𝐴‖1= max𝑗∑𝑛𝑖=1|𝑎𝑖𝑗|,

‖𝐴‖∞= max𝑖∑𝑛𝑗=1|𝑎𝑖𝑗|.

For any 𝐴 = [𝑎𝑖𝑗]𝜖𝑀{𝑚,𝑛}(𝐶), the Frobenious (or Euclidean) norm of matrix A is

‖𝐴‖𝐸= (∑ ∑ |𝑎𝑖𝑗| 2 𝑚 𝐽=1 𝑛 𝑖=1 ) 1/2 , (5)

and the spectral norm of matrix A is

‖𝐴‖2= √_{1≤𝑖≤𝑛}𝑚𝑎𝑥 𝜆𝑖(𝐴𝐻𝐴) , (6)

where 𝜆𝑖(𝐴𝐻𝐴) is an eigenvalue of 𝐴𝐻𝐴, and 𝐴𝐻 is the conjugate transpose of matrix A.

The maximum column length norm c

₁

₁

𝑐1(𝐴) = max𝑗√∑ |𝑎𝑖𝑗| 2 𝑛 𝑖=1 , 𝑟1(𝐴) = max𝑖√∑ |𝑎𝑖𝑗| 2 𝑚 𝑗=1 .

For any 𝐴, 𝐵𝜖𝑀{𝑚,𝑛}(𝐶), the Hadamard product of A, B is entrywise product and defined by [3,4]

𝐴 ∘ 𝐵 = (𝑎𝑖𝑗𝑏𝑖𝑗)

and have the following properties

‖𝐴 ∘ 𝐵‖2≤ 𝑟1(𝐴)𝑐1(𝐵), ‖𝐴 ∘ 𝐵‖ ≤ ‖𝐴‖ ‖𝐵‖. (7)

Let 𝐴 𝜖 𝑀_{𝑚,𝑛}(𝐶), and 𝐵 𝜖 𝑀_{𝑝,𝑞}(𝐶) be given, then the Kronecker product of A, B is defined by [4,6,8]

Uygun, Ş. and Aytar, H., Journal of Scientific Reports-A, Number 45, 90-100, December 2020.

93 ‖𝐴 ⨂ 𝐵‖ = [𝑎11⋮𝐵 ⋯ 𝑎⋱ 1𝑛⋮𝐵

𝑎𝑚1𝐵 ⋯ 𝑎𝑚𝑛𝐵

] and have the following properties

‖𝐴 ⨂ 𝐵‖2= ‖𝐴‖2 ‖𝐵‖2,

Let 𝐴 𝜖 𝑀{𝑚,𝑛}(𝐶) be given, then the inequality is hold [1,2] 1

√𝑛 ‖𝐴‖𝐸≤ ‖𝐴‖2 ≤ ‖𝐴‖𝐸. (9)

2. SOME SUM FORMULAS FOR k-JACOBSTHAL AND k-JACOBSTHAL LUCAS NUMBERS

Let 𝑘 ≠ −1, 1. The summation formulas for the k-Jacobsthal and k-Jacobsthal Lucas sequences are ∑𝑛−1𝑖=0𝑗𝑘,𝑖 = 𝑗_𝑘,𝑛+2𝑗_{𝑘,𝑛−1}−1 𝑘+1 , (10) ∑𝑛−1𝑖=0𝑐𝑘,𝑖= 𝑐_𝑘,𝑛+2 𝑐_{𝑘,𝑛−1}+𝑘−2 𝑘+1 . (11)

The summation of the squares of k-Jacobsthal sequence and k-Jacobsthal Lucas sequence are ∑𝑛𝑖=1𝑗𝑘,𝑖2 = 1 𝑘2+8[ 4𝑐𝑘,2𝑛−𝑐𝑘,2𝑛+2−𝑐𝑘,2+2 5−𝑐_𝑘,2 + 2 (−1) 𝑛+1_𝑗 𝑛+1], (12) ∑𝑛𝑖=1𝑐𝑘,𝑖2 = 4𝑐_𝑘,2𝑛−𝑐_𝑘,2𝑛+2−𝑐_𝑘,2+2 5−𝑐_𝑘,2 − 2 (−1) 𝑛+1_𝑗 𝑛+1 . (13)

The summation of the squares of k-Jacobsthal and k-Jacobsthal Lucas sequence with negative indices are demonstrated ∑ (𝑗𝑘,−𝑖) 2 𝑛 𝑖=1 = ∑ ( 𝑗_𝑘,𝑖 2𝑖) 2 = 𝑛 𝑖=1 1 𝑘2₊₈[ 1 1−𝑘2 ( 𝑐_𝑘,2𝑛−𝑐_𝑘,2𝑛+2 22𝑛 − 𝑘2+ 4) − 𝑗_𝑛+1 2𝑛−1 ], (14) ∑ (𝑐𝑘,−𝑖) 2 𝑛 𝑖=1 = ∑ ( 𝑐_𝑘,𝑖 2𝑖) 2 =_1−𝑘1₂ 𝑛 𝑖=1 ( 𝑐_{𝑘,2𝑛−}𝑐_𝑘,2𝑛+2 22𝑛 + 4 − 𝑘2) + 𝑗𝑛+1 2𝑛−1− 4. (15)

Some summation formulas for k-Jacobsthal sequence and k-Jacobsthal Lucas sequence are as follows ∑𝑛−1𝑖=1𝑐𝑘,2𝑖= 4𝑐_{𝑘,2𝑛−2−}𝑐_𝑘,2𝑛+𝑘2−4 1−𝑘2 , (16) ∑ 𝑐𝑘,2𝑖 4𝑖 𝑛−1 𝑖=1 = 𝑐_{𝑘,2𝑛−2−}𝑐_𝑘,2𝑛 4𝑛−1_(1−𝑘2₎ + 𝑘2+2 1−𝑘2, (17) ∑ 𝑐𝑘,2𝑖+2 4𝑖 𝑛−1 𝑖=1 =𝑐₄𝑘,2𝑛−𝑛−1(1−𝑘𝑐𝑘,2𝑛+22) + 4 𝑘2+2 1−𝑘2, (18)

Uygun, Ş. and Aytar, H., Journal of Scientific Reports-A, Number 45, 90-100, December 2020. 94 ∑𝑛−1𝑖=1𝑐𝑘,2𝑖+2= 4𝑐_{𝑘,2𝑛−}𝑐_𝑘,2𝑛+2+𝑘2−4 1−𝑘2 − (𝑘2+ 4), (19) ∑ (−1)𝑖+1_𝑗 𝑖+1 𝑛−1 𝑖=1 =4(−1) 𝑛_{𝐽𝑛−1−𝑛+1} 3 , (20) ∑ 𝑗𝑖+1 2𝑖−1 𝑛−1 𝑖=1 = −_3.2𝑗𝑛−1𝑛−2+ 4𝑛−4 3 . (21)

3. LOWER AND UPPER BOUNDS OF TOEPLITZ MATRICES INVOLVING k-JACOBSTHAL NUMBERS

Theorem 1: Let 𝐴 = 𝑇 (𝑗_𝑘,0 , 𝑗_𝑘,1 , … , 𝑗_{𝑘,𝑛−1} ) be a Toeplitz matrix with k-Jacobsthal numbers, then

the largest absolute column sum (1-norm) and the largest absolute row sum (∞-norm) of A are ‖𝐴‖1= ‖𝐴‖∞=

(𝑘 + 2) 𝑗𝑘,𝑛+ 2𝑗𝑘,𝑛−1− 1

𝑘 + 1 .

Proof. Clearly, the explicit form of this matrix as follows:

𝐴 = [ 𝑗𝑘,0 𝑗𝑘,−1 𝑗𝑘,−2 ⋯ 𝑗𝑘,1−𝑛 𝑗𝑘,1 𝑗𝑘,0 𝑗𝑘,−1 ⋯ 𝑗𝑘,2−𝑛 𝑗𝑘,2 𝑗𝑘,1 𝑗𝑘,0 ⋯ 𝑗𝑘,3−𝑛 ⋮ ⋮ ⋮ ⋱ ⋮ 𝑗𝑘,𝑛−1 𝑗𝑘,𝑛−2 𝑗𝑘,𝑛−3 ⋯ 𝑗𝑘,0 ] (22)

By the definitions of 1-norm and ∞-norm, and (10), it is easily seen ‖𝐴‖1= max 𝑗 ∑|𝑎𝑖𝑗| 𝑛 𝑖=1 = ∑|𝑎𝑖1| 𝑛 𝑖=1 = ∑ 𝑗𝑘,𝑖 𝑛−1 𝑖=0 =𝑗𝑘,𝑛+ 2𝑗𝑘,𝑛−1− 1 𝑘 + 1 , ‖𝐴‖∞= max𝑖∑𝑛𝑗=1|𝑎𝑖𝑗|= ∑𝑛𝑗=1|𝑎𝑛𝑗|= ∑𝑛−1𝑖=0 𝑗𝑘,𝑖= 𝑗_𝑘,𝑛+2𝑗_{𝑘,𝑛−1}−1 𝑘+1 . ∎

Theorem 2: Let 𝐴 = 𝑇 (𝑗𝑘,0 , 𝑗𝑘,1 , … , 𝑗𝑘,𝑛−1 ) be a Toeplitz matrix, then the Frobenious (or

Euclidean) norm of matrix A is

‖𝐴‖𝐸= √ 1 (𝑘2_+8)(1−𝑘2)2[𝑐𝑘,2𝑛+2+ 𝑐_𝑘,2𝑛+2 22𝑛−2 − 8𝑐𝑘,2𝑛− 2𝑐_𝑘,2𝑛 22𝑛−2+ 16𝑐𝑘,2𝑛−2+ 𝑐_{𝑘,2𝑛−2} 22𝑛−2 − 18] +(𝑛−1)(2𝑘_{(𝑘2+8)(𝑘}2−2)−4−𝑘₂₋₁₎ 2+_3(𝑘1₂₊₈₎[ 8(−1)𝑛_𝑗 𝑛−1−4 𝑗₂𝑛𝑛− 6𝑛 + 8] . (23)

Uygun, Ş. and Aytar, H., Journal of Scientific Reports-A, Number 45, 90-100, December 2020. 95 ‖𝐴‖𝐸2= ∑ ∑ 𝑎𝑖𝑗2 = 𝑛𝑗𝑘,02 𝑛 𝑗=1 + ∑ (𝑛 − 𝑚)𝑗𝑘,𝑚2+ ∑ (𝑛 − 𝑚)𝑗𝑘,−𝑚2 𝑛−1 𝑚=1 𝑛−1 𝑚=1 𝑛 𝑖=1 = ∑ ∑ 𝑗𝑘,𝑖2 𝑚 𝑖=1 𝑛−1 𝑚=1 + ∑ ∑ (𝑗𝑘,𝑖 2𝑖) 2 𝑚 𝑖=1 𝑛−1 𝑚=1 = 1 (𝑘2_{+ 8)}∑ [ 4𝑐𝑘,2𝑚− 𝑐𝑘,2𝑚+2− 𝑘2− 2 1 − 𝑘2 + 2 (−1)𝑚+1𝑗𝑚+1] 𝑛−1 𝑚=1 + 1 (𝑘2_{+ 8)}∑ 1 1 − 𝑘2( 𝑐𝑘,2𝑚− 𝑐𝑘,2𝑚+2 22𝑚 − 𝑘2+ 4) − 𝑗𝑚+1 2𝑚−1 𝑛−1 𝑚=1

And then by using the sum formulas (16), (19), (21), the following result is obtained:

= 1 (𝑘2_{+ 8)}[ 16𝑐𝑘,2𝑛−2− 4𝑐𝑘,2𝑛+ 4𝑘2− 16 (1 − 𝑘2₎2 + 𝑐𝑘,2𝑛+2− 4𝑐𝑘,2𝑛− 𝑘2+ 4 (1 − 𝑘2₎2 − (𝑘2_{+ 4)} 𝑘2_{− 1} +(𝑘 2_{+ 2)(𝑛 − 1)} 𝑘2_{− 1} + 8(−1)𝑛_𝑗 𝑛−1− 2𝑛 + 2 3 ] + 1 (𝑘2_{+ 8)} { 1 (1 − 𝑘2₎2[ −𝑐𝑘,2𝑛+ 𝑐𝑘,2𝑛−2 22𝑛−2 + 2 + 𝑘2+ −𝑐𝑘,2𝑛+ 𝑐𝑘,2𝑛+2 22𝑛−2 − 8 − 4𝑘2] + (𝑛 − 1) [𝑘2− 4 𝑘2_{− 1}] + 𝑗𝑛−1 3. 2𝑛−2− 4𝑛 − 4 3 } = 1 (𝑘2_{+ 8)(1 − 𝑘}2₎2[𝑐𝑘,2𝑛+2+ 𝑐𝑘,2𝑛+2 22𝑛−2 − 8𝑐𝑘,2𝑛− 2𝑐𝑘,2𝑛 22𝑛−2+ 16𝑐𝑘,2𝑛−2+ 𝑐𝑘,2𝑛−2 22𝑛−2 − 10] +(𝑛−1)(2𝑘_{(𝑘2+8)(𝑘}2−2)−4−𝑘₂₋₁₎ 2+_3(𝑘1₂₊₈₎[ 8(−1)𝑛_𝑗 𝑛−1+ 𝑗₂𝑛−2𝑛−1− 6𝑛 + 6]. ∎

Theorem 3: Let 𝐴 = 𝑇 (𝑗_𝑘,0 , 𝑗_𝑘,1 , … , 𝑗_{𝑘,𝑛−1} ) be Toeplitz matrix, then the lower and upper bounds for

the spectral norm of A are obtained as

√ 1 𝑛(𝑘2_+8)(1−𝑘2)2[𝑐𝑘,2𝑛+2+ 𝑐𝑘,2𝑛+2 22𝑛−2 − 8𝑐𝑘,2𝑛− 2𝑐𝑘,2𝑛 22𝑛−2+ 16𝑐𝑘,2𝑛−2+ 𝑐𝑘,2𝑛−2 22𝑛−2 − 10] +(𝑛−1)(2𝑘_{𝑛(𝑘2+8)(𝑘}2−2)−4−𝑘₂₋₁₎ 2+_3𝑛(𝑘1₂₊₈₎[ 8(−1)𝑛_𝑗 𝑛−1+ 𝑗₂𝑛−1𝑛−2− 6𝑛 + 6] ≤ ‖𝐴‖2 (24) ‖𝐴‖2 ≤ √ [ _𝑘21₊₈(4𝑐𝑘,2𝑛−4−𝑐𝑘,2𝑛−2−𝑘2−2 1−𝑘2 + 2(−1)𝑛−1𝑗𝑛−1) + 1] [ _𝑘21₊₈(4𝑐𝑘,2𝑛−2−𝑐𝑘,2𝑛−𝑘2−2 1−𝑘2 + 2(−1)𝑛𝑗𝑛) ] (25)

Proof. By (23) and using the property (9), the left hand side of the inequality is completed.

On the other hand, let 𝐴 = 𝐵 ∘ 𝐶 whereas 𝐵 = 𝑏𝑖𝑗= {

𝑏𝑖𝑗= 1 𝑗 = 1

𝑏𝑖𝑗 = 𝑗𝑘,𝑖−𝑗 𝑗 ≠ 1 and 𝐶 = 𝑐𝑖𝑗 = {

𝑐𝑖𝑗 = 𝑗𝑘,𝑖−𝑗 𝑗 = 1

Uygun, Ş. and Aytar, H., Journal of Scientific Reports-A, Number 45, 90-100, December 2020. 96 𝐵 = [ 1 𝑗𝑘,−1 𝑗𝑘,−2 ⋯ 𝑗𝑘,1−𝑛 1 𝑗𝑘,0 𝑗𝑘,−1 ⋯ 𝑗𝑘,2−𝑛 1 𝑗𝑘,1 𝑗𝑘,0 ⋯ 𝑗𝑘,3−𝑛 ⋮ ⋮ ⋮ ⋱ ⋮ 1 𝑗𝑘,𝑛−2 𝑗𝑘,𝑛−3 ⋯ 𝑗𝑘,0 ] and 𝐶 = [ 𝑗𝑘,0 1 1 ⋯ 1 𝑗𝑘,1 1 1 ⋯ 1 𝑗𝑘,2 1 1 ⋯ 1 ⋮ ⋮ ⋮ ⋱ ⋮ 𝑗𝑘,𝑛−1 1 1 ⋯ 1] 𝑟1(𝐵) = max_{1≤𝑖≤𝑛}√∑|𝑏𝑖𝑗| 2 𝑛 𝑗=1 = √∑|𝑏𝑛𝑗| 2 𝑛 𝑗=1 = √∑ 𝑗𝑘,𝑖2+ 1 𝑛−2 𝑖=0 = √ 1 𝑘2_{+ 8}( 4𝑐𝑘,2𝑛−4− 𝑐𝑘,2𝑛−2− 𝑘2− 2 1 − 𝑘2 + 2(−1)𝑛−1𝑗𝑛−1) + 1 𝑐1(𝐶) = max_{1≤𝑖≤𝑛}√∑|𝑐𝑖𝑗| 2 𝑛 𝑖=1 = √∑|𝑐𝑗1| 2 𝑛 𝑖=1 = √∑ 𝑗𝑘,𝑖2 𝑛−1 𝑖=0 = √ _𝑘21₊₈(4𝑐𝑘,2𝑛−2−𝑐𝑘,2𝑛−𝑘2−2 1−𝑘2 + 2(−1)𝑛𝑗𝑛) .

By using the property (7), the right hand side of the inequality is completed:

𝑟1(𝐵)𝑐1(𝐶) = √ [ _𝑘21₊₈(4𝑐𝑘,2𝑛−4−𝑐𝑘,2𝑛−2−𝑘2−2 1−𝑘2 + 2(−1)𝑛−1𝑗𝑛−1) + 1] [ _𝑘21₊₈( 4𝑐𝑘,2𝑛−2−𝑐𝑘,2𝑛−𝑘2−2 1−𝑘2 + 2(−1)𝑛𝑗𝑛) ] ∎

4. LOWER AND UPPER BOUNDS OF TOEPLITZ MATRICES INVOLVING k-JACOBSTHAL LUCAS NUMBERS

Theorem 4: Let the elements of the Toeplitz matrix be k-Jacobsthal Lucas numbers

𝐴 = 𝑇 (𝑐𝑘,0 , 𝑐𝑘,1 , … , 𝑐𝑘,𝑛−1 ), then 1-norm and ∞-norm of A are

‖𝐴‖1= ‖𝐴‖∞=(𝑘+2) 𝑗𝑘,𝑛_𝑘+1+2𝑗𝑘,𝑛−1−1.

Proof. Clearly, the explicit form of this matrix as follows:

𝐴 = [ 𝑐𝑘,0 𝑐𝑘,−1 𝑐𝑘,−2 ⋯ 𝑐𝑘,1−𝑛 𝑐𝑘,1 𝑐𝑘,0 𝑐𝑘,−1 ⋯ 𝑐𝑘,2−𝑛 𝑐𝑘,2 𝑐𝑘,1 𝑐𝑘,0 ⋯ 𝑐𝑘,3−𝑛 ⋮ ⋮ ⋮ ⋱ ⋮ 𝑐𝑘,𝑛−1 𝑐𝑘,𝑛−2 𝑐𝑘,𝑛−3 ⋯ 𝑐𝑘,0 ] . (26)

Uygun, Ş. and Aytar, H., Journal of Scientific Reports-A, Number 45, 90-100, December 2020. 97 ‖𝐴‖1= max𝑗∑𝑛𝑖=1|𝑎𝑖𝑗|= ∑𝑖=1𝑛 |𝑎𝑖1|= ∑𝑛−1𝑖=0 𝑐𝑘,𝑖 = 𝑐𝑘,𝑛+2 𝑐_𝑘+1𝑘,𝑛−1+𝑘−2, ‖𝐴‖∞= max_𝑖 ∑|𝑎𝑖𝑗| 𝑛 𝑗=1 = ∑|𝑎𝑛𝑗| 𝑛 𝑗=1 = ∑ 𝑐𝑘,𝑖 𝑛−1 𝑖=0 = 𝑐𝑘,𝑛+ 2 𝑐𝑘,𝑛−1+ 𝑘 − 2 𝑘 + 1 .

In the following theorem, we give the Euclidean (Frobenius) norm of the matrix involving k-Jacobsthal Lucas numbers.

Theorem 5: Let 𝐴 = 𝑇 (𝑐_𝑘,0 , 𝑐_𝑘,1 , … , 𝑐_{𝑘,𝑛−1} ) be Toeplitz matrix with k-Jacobsthal Lucas numbers,

then the Frobenious (or Euclidean) norm of matrix A is

‖𝐴‖𝐸= √ 16𝑐_{𝑘,2𝑛−2}−8𝑐_𝑘,2𝑛+𝑐_𝑘,2𝑛+2−𝑘2₋₂₀ (1−𝑘2)2 + 𝑐_{𝑘,2𝑛−2}−2𝑐_𝑘,2𝑛+𝑐_𝑘,2𝑛+2 22𝑛−2_(1−𝑘2)2 +(𝑘2+4)+(𝑛−1)(6𝑘_1−𝑘2 2−6) +−8(−1)𝑛𝑗𝑛−1+6𝑛−8 3 + 𝑗_𝑛 3.2𝑛−2. (27)

Proof. Let A be an n×n matrix as in (26). Then by the definition of Frobenius norm and by (5), (13),

(15), we can obtain the following equations for matrix A

‖𝐴‖𝐸2= ∑ ∑ 𝑎𝑖𝑗2 = 𝑛𝑐𝑘,02 𝑛 𝑗=1 + ∑(𝑛 − 𝑖)𝑐𝑘,𝑖2+ ∑(𝑛 − 𝑖)𝑐𝑘,−𝑖2 𝑛−1 𝑖=1 𝑛−1 𝑖=1 𝑛 𝑖=1 = 4𝑛 + ∑ ∑ 𝑐𝑘,𝑖2+ ∑ ∑ ( 𝑐𝑘,𝑖 2𝑖) 2 𝑚 𝑖=1 𝑛−1 𝑚=1 𝑚 𝑖=1 𝑛−1 𝑚=1 = ∑4𝑐𝑘,2𝑚− 𝑐𝑘,2𝑚+2− 𝑘2− 4 1 − 𝑘2 − 2(−1)𝑚+1𝑗𝑚+1 𝑛−1 𝑚=1 + ∑𝑐𝑘,2𝑚− 𝑐𝑘,2𝑚+2 22𝑚_{(1 − 𝑘}2₎ + 3𝑘2 1 − 𝑘2+ 𝑗𝑚+1 2𝑚−1 𝑛−1 𝑚=1 =16𝑐𝑘,2𝑛−2− 8𝑐𝑘,2𝑛+ 𝑐𝑘,2𝑛+2− (𝑘 2_{− 4)}2 (1 − 𝑘2₎2 + 𝑐𝑘,2𝑛−2− 2𝑐𝑘,2𝑛+ 𝑐𝑘,2𝑛+2 22𝑛−2_{(1 − 𝑘}2₎2 +(−4𝑘 2_{− 8) + (𝑛 − 1)(2𝑘}2_{− 4)} 1 − 𝑘2 + −8(−1)𝑛_𝑗 𝑛−1+ 6𝑛 − 6 3 − 𝑗𝑛−1 3. 2𝑛−2. ∎

Theorem 6: Let 𝐴 = 𝑇 (𝑐𝑘,0 , 𝑐𝑘,1 , … , 𝑐𝑘,𝑛−1) be Toeplitz matrix, then the lower and upper bounds

for the spectral norm of A are obtained as

√ 16𝑐𝑘,2𝑛−2−8𝑐𝑘,2𝑛+𝑐𝑘,2𝑛+2−(𝑘2−4)2 𝑛(1−𝑘2)2 + 𝑐𝑘,2𝑛−2−2𝑐𝑘,2𝑛+𝑐𝑘,2𝑛+2 22𝑛−2_{𝑛(1−𝑘}2)2 +(−4𝑘2−8)+(𝑛−1)(2𝑘_{(1−𝑘2)𝑛} 2−4) +−8(−1)𝑛𝑗𝑛−1+6𝑛−6 3𝑛 − 𝑗𝑛−1 3𝑛.2𝑛−2 ≤ ‖𝐴‖2 (28) ‖𝐴‖2≤ √ 4 𝑐_{𝑘,2𝑛−4}−𝑐_{𝑘,2𝑛−2}−𝑘2₋₂ 1−𝑘2 − 2(−1)𝑛−1 𝑗𝑛−1+ 5 4 𝑐𝑘,2𝑛−2−𝑐𝑘,2𝑛−𝑘2−2 1−𝑘2 − 2(−1)𝑛 𝑗𝑛+ 4 (29)

Uygun, Ş. and Aytar, H., Journal of Scientific Reports-A, Number 45, 90-100, December 2020.

98

Proof. By (27), and using the property (9), the left hand side of the inequality is completed. On the

other hand, let 𝐴 = 𝐵 ∘ 𝐶 whereas 𝐵 = 𝑏𝑖𝑗= { 𝑏𝑖𝑗 = 1 𝑗 = 1 𝑏𝑖𝑗 = 𝑐𝑘,𝑖−𝑗 𝑗 ≠ 1 and 𝐶 = 𝑐𝑖𝑗 = { 𝑐𝑖𝑗 = 𝑐𝑘,𝑖−𝑗 𝑗 = 1 𝑐𝑖𝑗= 1 𝑗 ≠ 1 𝐵 = [ 1 𝑐𝑘,−1 𝑐𝑘,−2 ⋯ 𝑐𝑘,1−𝑛 1 𝑐𝑘,0 𝑐𝑘,−1 ⋯ 𝑐𝑘,2−𝑛 1 𝑐𝑘,1 𝑐𝑘,0 ⋯ 𝑐𝑘,3−𝑛 ⋮ ⋮ ⋮ ⋱ ⋮ 1 𝑐𝑘,𝑛−2 𝑐𝑘,𝑛−3 ⋯ 𝑐𝑘,0 ] and 𝐶 = [ 𝑐𝑘,0 1 1 ⋯ 1 𝑐𝑘,1 1 1 ⋯ 1 𝑐𝑘,2 1 1 ⋯ 1 ⋮ ⋮ ⋮ ⋱ ⋮ 𝑐𝑘,𝑛−1 1 1 ⋯ 1] .

Then, by using the sum formula (13) and the definition of the maximum column length norm and the maximum row length norm, the following equalities are hold:

𝑟1(𝐵) = max 1≤𝑖≤𝑛√∑|𝑏𝑖𝑗| 2 𝑛 𝑗=1 = √∑|𝑏𝑛𝑗| 2 𝑛 𝑗=1 = √∑ 𝑐𝑘,𝑖2+ 1 𝑛−2 𝑖=0 = √4 𝑐𝑘,2𝑛−4−𝑐𝑘,2𝑛−2−𝑘2−2 1−𝑘2 − 2(−1)𝑛−1 𝑗𝑛−1+ 5, 𝑐1(𝐶) = max_{1≤𝑖≤𝑛}√∑|𝑐𝑖𝑗| 2 𝑛 𝑖=1 = √∑|𝑐𝑗1| 2 𝑛 𝑖=1 = √∑ 𝑐𝑘,𝑖2 𝑛−1 𝑖=0 = √4 𝑐𝑘,2𝑛−2−𝑐𝑘,2𝑛−𝑘2−2 1−𝑘2 − 2(−1)𝑛 𝑗𝑛+ 4.

If we use the equations given in (7), the right hand side of the inequality is completed. ∎

Corollary 7: Let 𝐴 = 𝑇 (𝑗𝑘,0 , 𝑗𝑘,1 , … , 𝑗𝑘,𝑛−1 ) and 𝐵 = 𝑇 (𝑐𝑘,0 , 𝑐𝑘,1 , … , 𝑐𝑘,𝑛−1) be Toeplitz matrix

with k-Jacobsthal and the k-Jacobsthal Lucas numbers, then the Euclidean norm of Kronecker product of these matrices is given as:

‖𝐴 ⨂ 𝐵‖𝐸= √ 1 (𝑘2_+8)(1−𝑘2)2[𝑐𝑘,2𝑛+2+ 𝑐_𝑘,2𝑛+2 22𝑛−2 − 8𝑐𝑘,2𝑛− 2𝑐_𝑘,2𝑛 22𝑛−2+ 16𝑐𝑘,2𝑛−2+ 𝑐_{𝑘,2𝑛−2} 22𝑛−2 − 18] +(𝑛−1)(2𝑘_{(𝑘2+8)(𝑘}2−2)−4−𝑘₂₋₁₎ 2+_3(𝑘21₊₈₎[ 8(−1)𝑛_𝑗 𝑛−1−4 𝑗₂𝑛𝑛− 6𝑛 + 8] ⋅ √ 16𝑐_{𝑘,2𝑛−2}−8𝑐_𝑘,2𝑛+𝑐_𝑘,2𝑛+2−𝑘2₋₂₀ (1−𝑘2)2 + 𝑐_{𝑘,2𝑛−2}−2𝑐_𝑘,2𝑛+𝑐_𝑘,2𝑛+2 22𝑛−2_(1−𝑘2)2 (𝑘2_{+4)+(𝑛−1)(6𝑘}2₋₆₎ 1−𝑘2 + −8(−1)𝑛_{𝑗𝑛−1+6𝑛−8} 3 + 𝑗_𝑛 3.2𝑛−2

Uygun, Ş. and Aytar, H., Journal of Scientific Reports-A, Number 45, 90-100, December 2020.

99

Corollary 8: Let 𝐴 = 𝑇 (𝑗_𝑘,0 , 𝑗_𝑘,1 , … , 𝑗_{𝑘,𝑛−1} ) and 𝐵 = 𝑇 (𝑐_𝑘,0 , 𝑐_𝑘,1 , … , 𝑐_{𝑘,𝑛−1}) be Toeplitz matrix

with k-Jacobsthal and the k-Jacobsthal Lucas numbers, then the upper bound for the spectral norm of Kronecker product of these matrices is given as:

‖𝐴 ⨂ 𝐵‖2≤ √ [ _𝑘21₊₈(4𝑐𝑘,2𝑛−4−𝑐𝑘,2𝑛−2−𝑘2−2 1−𝑘2 + 2(−1)𝑛−1𝑗𝑛−1) + 1] [ _𝑘21₊₈(4𝑐𝑘,2𝑛−2−𝑐𝑘,2𝑛−𝑘2−2 1−𝑘2 + 2(−1)𝑛𝑗𝑛) ] ∙ √ 4 𝑐_{𝑘,2𝑛−4}−𝑐_{𝑘,2𝑛−2}−𝑘2₋₂ 1−𝑘2 − 2(−1)𝑛−1 𝑗𝑛−1+ 5 4 𝑐_{𝑘,2𝑛−2}−𝑐_𝑘,2𝑛−𝑘2₋₂ 1−𝑘2 − 2(−1)𝑛 𝑗𝑛+ 4

Proof. The proof is seen easily by ‖𝐴 ⨂𝐵 ‖2= ‖𝐴‖2 ‖𝐵‖2 and (25), (29). ∎

Corollary 9 : Let 𝐴 = 𝑇 (𝑗𝑘,0 , 𝑗𝑘,1 , … , 𝑗𝑘,𝑛−1 ) and 𝐵 = 𝑇 (𝑐𝑘,0 , 𝑐𝑘,1 , … , 𝑐𝑘,𝑛−1) be Toeplitz matrix

with k-Jacobsthal and the k -Jacobsthal Lucas numbers, then the upper bound for the spectral norm of Hadamard product of the matrices is

‖𝐴 ∘ 𝐵 ‖2≤ √ [ _𝑘21₊₈(4𝑐𝑘,2𝑛−4−𝑐𝑘,2𝑛−2−𝑘2−2 1−𝑘2 + 2(−1)𝑛−1𝑗𝑛−1) + 1] [ _𝑘21₊₈(4𝑐𝑘,2𝑛−2−𝑐𝑘,2𝑛−𝑘2−2 1−𝑘2 + 2(−1)𝑛𝑗𝑛) ] ∙ √( 4 𝑐_{𝑘,2𝑛−4}−𝑐_{𝑘,2𝑛−2}−𝑘2₋₂ 1−𝑘2 − 2(−1)𝑛−1 𝑗𝑛−1+ 5) (4 𝑐𝑘,2𝑛−2−𝑐𝑘,2𝑛−𝑘2−2 1−𝑘2 − 2(−1)𝑛 𝑗𝑛+ 4)

Proof. The proof is seen easily by ‖𝐴 ∘ 𝐵 ‖₂≤ ‖𝐴‖₂ ‖𝐵‖₂ and (25), (29). ∎

Corollary 10 : Let 𝐴 = 𝑇 (𝑗𝑘,0 , 𝑗𝑘,1 , … , 𝑗𝑘,𝑛−1 ) and 𝐵 = 𝑇 (𝑐𝑘,0 , 𝑐𝑘,1 , … , 𝑐𝑘,𝑛−1) be Toeplitz

matrix with k-Jacobsthal and the k -Jacobsthal Lucas numbers, then the upper bound for the Euclid norm of Hadamard product of the matrices is

‖𝐴 ∘ 𝐵 ‖𝐸≤ √ 1 (𝑘2_{+ 8)(1 − 𝑘}2₎2[𝑐𝑘,2𝑛+2+ 𝑐𝑘,2𝑛+2 22𝑛−2 − 8𝑐𝑘,2𝑛− 2𝑐𝑘,2𝑛 22𝑛−2+ 16𝑐𝑘,2𝑛−2+ 𝑐𝑘,2𝑛−2 22𝑛−2 − 18] +(𝑛 − 1)(2𝑘2− 2) − 4 − 𝑘2 (𝑘2_{+ 8)(𝑘}2_{− 1)} + 1 3(𝑘2_{+ 8)}[ 8(−1)𝑛𝑗𝑛−1− 4 𝑗𝑛 2𝑛 − 6𝑛 + 8] . ∙ √ 16𝑐𝑘,2𝑛−2− 8𝑐𝑘,2𝑛+ 𝑐𝑘,2𝑛+2− 𝑘2− 20 (1 − 𝑘2₎2 + 𝑐𝑘,2𝑛−2− 2𝑐𝑘,2𝑛+ 𝑐𝑘,2𝑛+2 22𝑛−2_{(1 − 𝑘}2₎2 +(𝑘2+ 4) + (𝑛 − 1)(6𝑘2− 6) 1 − 𝑘2 + 8(−1)𝑛+1_𝑗 𝑛−1+ 6𝑛 − 8 3 + 𝑗𝑛 3. 2𝑛−2

Uygun, Ş. and Aytar, H., Journal of Scientific Reports-A, Number 45, 90-100, December 2020.

100

REFERENCES

[1] Mathias, R., (1990), The spectral norm of nonnegative matrix, Linear Algebra and its Applications, 131, 269-284.

[2] Zielke, G., (1988), Some remarks on matrix norms, condition numbers and error estimates for linear equations, Linear Algebra and its Applications, 110, 29-41.

[3] Reams, R., (1999), Hadamard inverses square roots and products of almost semi-definite matrices, Linear Algebra and its Applications, 288, 35-43.

[4] Horn, R. A., Johnson, C. R., (1991), Topics in matrix analysis, Cambridge University Press, Cambridge.

[5] Visick, G., (2000), A quantitative version of the observation that the Hadamard product is a principal submatrix of the Kronecker product, Linear Algebra Appl., 304, 45-68.

[6] Solak, S., (2005), On the norms of circulant matrices with the Fibonacci and Lucas numbers, Appl. Math. Comput., 160, 125-132.

[7] Akbulak, M., Bozkurt, D., (2008), On the norms of Toeplitz matrices involving Fibonacci and Lucas numbers, Hacettepe Journal of Mathematics and Statistics, 37(2), 89-95.

[8] Shen, S., (2012), On the Norms of Toeplitz Matrices Involving k-Fibonacci and k-Lucas Numbers, Int. J. Contemp. Math. Sciences, 7(8), 363-368.

[9] Daşdemir, A., (2016), On the norms of Toeplitz Matrices with the Pell, Pell-Lucas and Modified Pell numbers, Journal of Engineering Technology and Applied Sciences, 1(2), 51-57.

[10] Kocer, E. G., (2007), Circulant, Negacyclic and Semicirculant matrices with the modified Pell, Jacobsthal and Jacobsthal- Lucas numbers, Hacettepe Journal of Mathematics and Statistics, 36(2), 133-142.

[11] Raza, Z., Ali, M.A., (2015), On the Norms of Some Special Matrices with Generalized Fibonacci Sequence, J. Appl. Math. & Informatics, 33(5–6), 593–605.

[12] Uygun, S., Eldoğan, H., (2016), The k -Jacobsthal and k -Jacobsthal Lucas sequences, General Mathematics Notes, 36(1), 34-47.

[13] Uygun, ¸S. , (2016), Some Bounds for the Norms of Circulant Matrices with the k -Jacobsthal and k -Jacobsthal Lucas Numbers, Journal of Mathematics Research, 8(6), 133-138.