Response to Letter to the Editor: Clinical Implication of Atrial Conductance Impairment on New-Onset Atrial Fibrillation after Transcatheter Aortic Valve Implantation

KARAKULAK U. N.

ACTA CARDIOLOGICA SINICA, vol.40, no.6, pp.822-823, 2024 (SCI-Expanded, Scopus) identifier identifier identifier identifier

  • Publication Type: Article / Letter
  • Volume: 40 Issue: 6
  • Publication Date: 2024
  • Doi Number: 10.6515/acs.202411_40(6).20240919a
  • Journal Name: ACTA CARDIOLOGICA SINICA
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, CAB Abstracts, EMBASE, Veterinary Science Database
  • Page Numbers: pp.822-823
  • Hacettepe University Affiliated: Yes

Abstract

Characterizing the $6\times 6$ complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. In this paper, we investigate the eigenvalues and eigenvectors of CHMs. We show that any $n\times n$ CHM with dephased form has two constant eigenvalues $\pm\sqrt{n}$ and has two constant eigenvectors. We obtain the maximum numbers of identical eigenvalues of $6\times 6$ CHMs with dephased form and we extend this result to arbitrary dimension. We also show that there is no $6\times 6$ CHM with four identical eigenvalues. We conjecture that the eigenvalues and eigenvectors of $6\times 6$ CHMs will lead to the complete classification of $6\times 6$ CHMs.