Daniel Uzcátegui Contreras ; Dardo Goyeneche ; Ondřej Turek ; Zuzana Václavíková - Circulant matrices with orthogonal rows and off-diagonal entries of absolute value 1

cm:9524 - Communications in Mathematics, April 30, 2021, Volume 29 (2021), Issue 1 (Special Issue: Ostrava Mathematical Seminar) - https://doi.org/10.2478/cm-2021-0005
Circulant matrices with orthogonal rows and off-diagonal entries of absolute value 1

Authors: Daniel Uzcátegui Contreras ; Dardo Goyeneche ; Ondřej Turek ; Zuzana Václavíková

It is known that a real symmetric circulant matrix with diagonal entries d ≥ 0, off-diagonal entries ±1 and orthogonal rows exists only of order 2d + 2 (and trivially of order 1) [Turek and Goyeneche 2019]. In this paper we consider a complex Hermitian analogy of those matrices. That is, we study the existence and construction of Hermitian circulant matrices having orthogonal rows, diagonal entries d ≥ 0 and any complex entries of absolute value 1 off the diagonal. As a particular case, we consider matrices whose off-diagonal entries are 4th roots of unity; we prove that the order of any such matrix with d different from an odd integer is n = 2d + 2. We also discuss a similar problem for symmetric circulant matrices defined over finite rings ℤ m . As an application of our results, we show a close connection to mutually unbiased bases, an important open problem in quantum information theory.


Volume: Volume 29 (2021), Issue 1 (Special Issue: Ostrava Mathematical Seminar)
Published on: April 30, 2021
Imported on: May 11, 2022
Keywords: General Mathematics,[MATH]Mathematics [math]


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