A note on the existence of BH(19,6) matrices

In this note we utilize a non-trivial block approach due to M. Petrescu to exhibit a Butson-type complex Hadamard matrix of order 19, composed of sixth roots of unity.Comment: 3 pages, preprin

Bilinear Forms on Finite Abelian Groups and Group-Invariant Butson Hadamard Matrices

Let $K$ be a finite abelian group and let $\exp(K)$ denote the least common multiple of the orders of the elements of $K$. A $BH(K,h)$ matrix is a $K$-invariant $|K|\times |K|$ matrix $H$ whose entries are complex $h$th roots of unity such that $HH^*=|K|I$, where $H^*$ denotes the complex conjugate transpose of $H$, and $I$ is the identity matrix of order $|K|$. Let $\nu_p(x)$ denote the $p$-adic valuation of the integer $x$. Using bilinear forms on $K$, we show that a $BH(K,h)$ exists whenever (i) $\nu_p(h) \geq \lceil \nu_p(\exp(K))/2 \rceil$ for every prime divisor $p$ of $|K|$ and (ii) $\nu_2(h) \ge 2$ if $\nu_2(|K|)$ is odd and $K$ has a direct factor $\mathbb{Z}_2$. Employing the field descent method, we prove that these conditions are necessary for the existence of a $BH(K,h)$ matrix in the case where $K$ is cyclic of prime power order

Diagonal Riccati Stability and Applications

We consider the question of diagonal Riccati stability for a pair of real matrices A, B. A necessary and sufficient condition for diagonal Riccati stability is derived and applications of this to two distinct cases are presented. We also describe some motivations for this question arising in the theory of generalised Lotka-Volterra systems