On the Frobenius Norm of Commutator of Cauchy-Hankel Matrix and Exchange Matrix

Year 2025, Volume: 17 Issue: 1, 26 - 32, 30.06.2025

Ahmet Öksüz , Süleyman Solak

https://doi.org/10.47000/tjmcs.1553098

Abstract

This paper gives upper and lower bounds for the Frobenius norm of the commutator of the exchange matrix and the Cauchy-Hankel matrix of the form $H_n=\left(\dfrac{2}{1+2(i+j)}\right)_{i,j=1}^n$.

Keywords

Cauchy-Hankel matrix , matrix commutator , frobenius norm.

References

• Audenaert, Koenraad M.R., Variance bounds, with an application to norm bounds for commutators, Linear Algebra and Its Applications, 432(2010), 1126–1143.
• Bozkurt, D., On ℓp norms of Cauchy-Toeplitz matrices, Linear and Multilinear Algebra, 44(1998), 341–346.
• Böttcher, A., Wenzel, D., How big can the commutator of two matrices be and how big is it typically?, Linear Algebra Appl., 403(2005), 216–228.
• Böttcher, A., Wenzel, D., The Frobenius norm and the commutator, Linear Algebra and Its Applications, 429(2008), 1864–1885.
• Chruscinski, D., Kimura, G., Ohno, H., Singal, T., One-parameter generalization of the B¨ottcher-Wenzel inequality and its application to open quantum dynamics, Linear Algebra and Its Applications, 656(2023), 158–166.
• Gil, M., A sharp bound for the Frobenius norm of self-commutators of matrices, Linear & Multilinear Algebra, 65(11)(2017), 2333–2339.
• Horn, R.A., & Johnson, C.R., Matrix Analysis, Cambridge University Press, 2012.
• Laszlo, L., Proof of B¨ottcher and Wenzel’s conjecture on commutator norms for 3-by-3 matrices, Linear Algebra Appl., 422(2007), 659–663.
• Lu, Z., Normal Scalar Curvature Conjecture and Its applications, Journal of Functional Analysis, 261(2011), 1284–1308.
• Moenck, R., On computing closed forms for summations, Proceedings of the MACSYMA User’s Conference, 225–236, 1977.
• Solak, S., Bahşi, M., On the Frobenius norm of commutator of Cauchy-Toeplitz matrix and exchange matrix, Turkish Journal of Mathematics, 47(5)(2023), 1550–1557.
• Solak, S., Bozkurt, D., On the spectral norms of Cauchy-Toeplitz and Cauchy-Hankel matrices, Applied Mathematics and Computations, 140(2003), 231–238.
• Solak, S., Bozkurt, D.,A note on bound for norms of Cauchy-Hankel matrices, Numerical Linear Algebra With Applications, 10(4)(2003), 377–382.
• Türkmen, R., Bozkurt, D., On the bounds for the norms of Cauchy-Toeplitz and Cauchy-Hankel matrices, Applied Mathematics and Computations, 132(2002), 633–642.
• Tyrtysnikov, E.E., Cauchy-Toeplitz matrices and some applications, Linear Algebra and its Applications, 149(1991), 1–18.
• Vong, S.W., Jin, X.Q., Proof of B¨ottcher and Wenzel’s conjecture, Oper. Matrices 2, (2008), 435–442.
• Wu, Y.D., Liu, X.Q., A short note on the Frobenius norm of the commutator, Mathematical Notes, 87(5/6)(2010), 903–907.