Circulant matrices with orthogonal rows and off-diagonal entries of absolute value 1

• Main Authors: Contreras Daniel Uzcátegui, Goyeneche Dardo, Turek Ondřej, Václavíková Zuzana
• Format: Article
• Language: English
• Published: Sciendo 2021-06-01
• Series: Communications in Mathematics
• Subjects: circulant matrix; orthogonal matrix; hadamard matrix; mutually unbiased base; 15b10; 15b36; 15b05
• Online Access: https://doi.org/10.2478/cm-2021-0005

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 ℤm. As an application of our results, we show a close connection to mutually unbiased bases, an important open problem in quantum information theory.