Circulant matrices with orthogonal rows and off-diagonal entries of absolute value 1 (Q2236370)

Let \(C\) be the circulant martix \[C=\begin{pmatrix} c_0&c_1&\cdots&c_{n-2}&c_{n-1}\\ c_{n-1}&c_0&c_1&&c_{n-2}\\ \vdots&c_{n-1}&c_0&\ddots&\vdots\\ c_2&&\ddots&\ddots&c_1\\ c_1&c_2&\cdots&c_{n-1}&c_0\\ \end{pmatrix} \] with generator \((c_0,c_1,\ldots,c_{n-1})\) satisfying \[ \begin{cases} c_0=d>0\\ |c_j|=1\quad\textrm{for all } j,\; 1\leqslant j\leqslant n-1\\ CC^*=(d^2+n-1)I.\tag{1} \end{cases} \] In [\textit{O. Turek} and \textit{D. Goyeneche}, Linear Algebra Appl. 569, 241--265 (2019; Zbl 1414.15039)] two of the authors of this paper studied the problem of determining the possible values of \(d\) for a fixed \(n\) when \(C\) is defined over \(\mathbb R\). In the present article they focus on the case when \(C\) is either Hermitian and defined over \(\mathbb C\) or symmetric and defined over \(\mathbb Z_m\). The authors compute the eigenvectors and the eigenvalues of the matrices. They study separately the cases when \(n\) is even and odd. If \(C\) is Hermitian and \(n\) is even they prove the following statements: (1) The number \(\sqrt{d^2+n-1}\) is an integer; (2) \(\sqrt{d^2+n-1}\) is a divisor of \(\frac n2\); (3) \(\sqrt{d^2+n-1}\) is a divisor of \(d^2-1\). In particular for odd \(d\) one has \(\sqrt{d^2+n-1}|\frac{d^2-1}2\); (4) If \(n-1\) is prime, then \(n=2d+2\). From these findings the authors derive a lot of consequences. In particular, it follows that there exists a Hermitian circulant Hadamard matrix of even order if and only if \(n\) is a square of an even integer. Another corollary states that if either \(\frac n2\) is prime or \(\frac n2\) is a product of two primes or \(\frac n2\) is a power of a prime, then \(n=2d+2\). The complex case when \(n\) is odd is more complicated. Thus, the authors present some partial results found by using some numerical methods. The section ends with the study of some special case, e.g., when the off-diagonal entries of \(C\) are from the set \(\left\{1,-1,i,-i\right\}\). In this case, it turns out that \(n=2d+2\), \(C\) is real, and the generator can take one of the forms \((d,-1,-1,\ldots,-1)\), \((d,1,-1,1,-1,\ldots,-1,1)\) when \(d\) is even, and only \((d,-1,-1,\ldots,-1)\) when \(d\) is a half-integer. If \(C\) is defined over \(\mathbb Z_m\) the third condition in (1) implies \[ 2dc_k+\sum_{\begin{smallmatrix}j=1\\ j\neq k\\ \end{smallmatrix}}c_j\cdot c_{n-k+j}\equiv 0\; (\mathrm{modulo}\: m) \] for all \(k\). Then, for any \(n\) and \(d\) so that \(n\equiv 2d+2\;\mathrm{modulo}\: m\) there exists a symmetric circulant matrix \(C\) satisfying (1). The paper ends with some applications, as for example the problem of mutually unbiased bases.