On the spectral norms of Cauchy–Toeplitz and Cauchy–Hankel matrices

Solak, Süleyman; Bozkurt, Durmuş

doi:10.1016/s0096-3003(02)00205-9

Cited by 19 publications

(8 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this paper, we give several necessary and sufficient conditions for an even order Cauchy tensor to be pos- However, there are still some questions that we are not sure now. The Cauchy matrix can be combined with many other structured matrices to form new structured matrices such as Cauchy-Toeplitz matrix and Cauchy-Hankel matrix [17,20,21]. Can we get the type of Cauchy-Toeplitz tensors and Cauchy-Hankel tensors?…”

Section: Final Remarksmentioning

confidence: 99%

Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors

Chen

2015

Journal of Industrial &Amp; Management Optimization

View full text Add to dashboard Cite

Motivated by symmetric Cauchy matrices, we define symmetric Cauchy tensors and their generating vectors in this paper. Hilbert tensors are symmetric Cauchy tensors. An even order symmetric Cauchy tensor is positive semi-definite if and only if its generating vector is positive. An even order symmetric Cauchy tensor is positive definite if and only if its generating vector has positive and mutually distinct entries. This extends Fiedler's result for symmetric Cauchy matrices to symmetric Cauchy tensors. Then, it is proven that the positive semi-definiteness character of an even order symmetric Cauchy tensor can be equivalently checked by the monotone increasing property of a homogeneous polynomial related to the Cauchy tensor. The homogeneous polynomial is strictly monotone increasing in the nonnegative orthant of the Euclidean space when the even order symmetric Cauchy tensor is positive definite. Furthermore, we prove that the Hadamard product of two positive semidefinite (positive definite respectively) symmetric Cauchy tensors is a positive semi-definite (positive definite respectively) tensor, which can be generalized to the Hadamard product of finitely many positive semi-definite (positive definite respectively) symmetric Cauchy tensors. At last, bounds of the largest H-eigenvalue of a positive semi-definite symmetric Cauchy tensor are given and several spectral properties on Z-eigenvalues of odd order symmetric Cauchy tensors are shown. Further questions on Cauchy tensors are raised.

show abstract

Section: Final Remarksmentioning

confidence: 99%

Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors

Chen

2015

Journal of Industrial &Amp; Management Optimization

View full text Add to dashboard Cite

show abstract

“…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 nxn 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 Eu-clidean norms.…”

Section: Introductionmentioning

confidence: 99%

On the Bounds for the Norms of R-Circulant Matrices With the Jacobsthal and Jacobsthal Lucas Numbers

Uygun¹

2017

Int. J. of Pure and Appl. Math.

View full text Add to dashboard Cite

show abstract

“…Lately, some authors studied the problems of the norms of some special matrices [11][12][13][14][15][16][17][18][19][20][21]. The author [11] found upper and lower bounds for the spectral norms of Toeplitz matrices such that ≡ − and − ≡ − .…”

Section: Introductionmentioning

confidence: 99%

“…, , −1 ), where { , } ∈ and { , } ∈ are -Fibonacci and -Lucas sequences, respectively, and they also give the bounds for the spectral norms of Kronecker and Hadamard products of these special matrices, respectively [14]. Solak and Bozkurt [16] have found out upper and lower bounds for the spectral norms of Cauchy-Toeplitz and Cauchy-Hankel matrices. Solak [18][19][20] ; then he has given some bounds for the and matrices concerned with the spectral and Euclidean norms.…”

Section: Introductionmentioning

confidence: 99%

Norms and Spread of the Fibonacci and Lucas RSFMLR Circulant Matrices

Jiang

2015

Abstract and Applied Analysis

View full text Add to dashboard Cite

Circulant type matrices have played an important role in networks engineering. In this paper, firstly, some bounds for the norms and spread of Fibonacci row skew first-minus-last right (RSFMLR) circulant matrices and Lucas row skew first-minus-last right (RSFMLR) circulant matrices are given. Furthermore, the spectral norm of Hadamard product of a Fibonacci RSFMLR circulant matrix and a Lucas RSFMLR circulant matrix is obtained. Finally, the Frobenius norm of Kronecker product of a Fibonacci RSFMLR circulant matrix and a Lucas RSFMLR circulant matrix is presented.

show abstract

On the spectral norms of Cauchy–Toeplitz and Cauchy–Hankel matrices

Cited by 19 publications

References 4 publications

Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors

Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors

On the Bounds for the Norms of R-Circulant Matrices With the Jacobsthal and Jacobsthal Lucas Numbers

Norms and Spread of the Fibonacci and Lucas RSFMLR Circulant Matrices

Contact Info

Product

Resources

About