Some Bounds on $\ell_p$ Matrix and $\ell_p$ Operator Norms of Almost Circulant, Cauchy-Toeplitz and Cauchy-Hankel Matrices

Solak, Süleyman; Bozkurt, Durmuş

doi:10.3390/mca7030211

Cited by 7 publications

(3 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the literature, there is an important class of structured matrices called Cauchy-Hankel matrices which is closely related with Cauchy matrices and Hankel matrices simultaneously [11,35,36]. A matrix A is called a Cauchy-Hankel matrices if it can be formulated as…”

Section: Properties Of Cauchy-hankel Tensorsmentioning

confidence: 99%

Further results on Cauchy tensors and Hankel tensors

Chen

2016

Applied Mathematics and Computation

View full text Add to dashboard Cite

In this article, we present various new results on Cauchy tensors and Hankel tensors. We first introduce the concept of generalized Cauchy tensors which extends Cauchy tensors in the current literature, and provide several conditions characterizing positive semi-definiteness of generalized Cauchy tensors with nonzero entries. As a consequence, we show that Cauchy tensors are positive semidefinite if and only if they are SOS (Sum-of-squares) tensors. Furthermore, we prove that all positive semi-definite Cauchy tensors are completely positive tensors, which means every positive semi-definite Cauchy tensor can be decomposed as the sum of nonnegative rank-1 tensors. We also establish that all the H-eigenvalues of nonnegative Cauchy tensors are nonnegative. Secondly, we present new mathematical properties of Hankel tensors. We prove that an even order Hankel tensor is Vandermonde positive semi-definite if and only if its associated plane tensor is positive semi-definite. We also show that, if the Vandermonde rank of a Hankel tensor A is less than the dimension of the underlying space, then positive semi-definiteness of A is equivalent to the fact that A is a complete Hankel tensor, and so, is further equivalent to the SOS property of A. Lastly, we introduce a new structured tensor called Cauchy-Hankel tensors, which is a special case of Cauchy tensors and Hankel tensors simultaneously. Sufficient and necessary conditions are established for an even order Cauchy-Hankel tensor to be positive definite. Final remarks are listed at the end of the paper.

show abstract

Section: Properties Of Cauchy-hankel Tensorsmentioning

confidence: 99%

Further results on Cauchy tensors and Hankel tensors

Chen

2016

Applied Mathematics and Computation

View full text Add to dashboard Cite

show abstract

“…Solak and Bozkurt [9] obtained upper bounds for the entrywise p-matrix norm and the matrix norms induced by the vector p-norm of almost circulant, Cauchy-Toepliz and Cauchy-Hankel matrices. Bani-Domi and Kittaneh [10] have established two general norm equalities for circulant and skew circulant operator matrices.…”

Section: Introduction and Historical Backgroundmentioning

confidence: 99%

The $p$-norm of circulant matrices

Bouthat,

Khare,

Mashreghi

et al. 2021

Preprint

View full text Add to dashboard Cite

In this note we study the induced p-norm of circulant matrices A(n, ±a, b), acting as operators on the Euclidean space R n . For circulant matrices whose entries are nonnegative real numbers, in particular for A(n, a, b), we provide an explicit formula for the p-norm, 1 ≤ p ≤ ∞. The calculation for A(n, −a, b) is more complex. The 2-norm is precisely determined. As for the other values of p, two different categories of upper and lower bounds are obtained. These bounds are optimal at the end points (i.e. p = 1 and p = ∞) as well as at p = 2.

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%

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

Some Bounds on \(\ell_p\) Matrix and \(\ell_p\) Operator Norms of Almost Circulant, Cauchy-Toeplitz and Cauchy-Hankel Matrices

Cited by 7 publications

References 0 publications

Further results on Cauchy tensors and Hankel tensors

Further results on Cauchy tensors and Hankel tensors

The $p$-norm of circulant matrices

Norms and Spread of the Fibonacci and Lucas RSFMLR Circulant Matrices

Contact Info

Product

Resources

About