turkmath.org

An nxn square matrix H is called Hadamard matrix if HH* = nI. Hadamard matrices have been used in many theoretical and engineering problems, some of them are Olivia MFSK (an amateur-radio digital protocol designed to work in difficult conditions on shortwave bands), balanced repeated replication (a technique used by statisticians to estimate the variance of a statistical estimator), coded aperture spectrometry (an instrument for measuring the spectrum of light), feedback delay networks (digital reverberation devices which use Hadamard matrices to blend sample values), Plackett–Burman design (the experiments for investigating the dependence of some measured quantity on a number of independent variables), robust parameter designs (investigating noise factor impacts on responses). If all the entries of an Hadamard matrix are powers of q-th root of unity for some integer q, then we call it Butson-Hadamard matrix. Butson-Hadamard matrices are used in many areas eg. communication systems and spectrometry. For instance, Fourier matrices are examples of such matrices. In this talk the general properties of Butson-Hadamard matrices will be covered and their applications will be illustrated. We will also present some new non-existence results on Butson-Hadamard matrices by using algebraic number theory.