# A note on the existence of matrices

Preprint

###### Abstract.

In this note we utilize a non-trivial block approach due to Petrescu [7] to exhibit a Butson-type complex Hadamard matrix of order , composed of sixth roots of unity.

2000 Mathematics Subject Classification. Primary 05B20, secondary 46L10.

Keywords and phrases. Complex Hadamard matrices, Butson-type, Petrescu’s array

## 1. A new Butson-type complex Hadamard matrix

A complex Hadamard matrix is an complex matrix with unimodular entries such that , where denotes the Hermitian adjoint and is the identity matrix of order . Throughout this note we are concerned with the special case when the entries are some th roots of unity. These matrices are called Butson-type complex Hadamard matrices and are denoted by [1]. The existence of matrices is wide open in general, but it is believed that real, i.e. matrices exist for every integral number . This is the famous Hadamard conjecture. Butson-type complex Hadamard matrices have applications in signal processing, coding theory [4] and harmonic analysis [6], among other things. They can also lead to constructions of real Hadamard matrices. This approach was demonstrated very recently in [3], where matrices were considered. For further results on matrices we refer the reader to Horadam’s celebrated book [5].

The main result of this note is the following

###### Theorem 1.1.

There exists a matrix.

Order was listed as the smallest outstanding order of matrices in [2]. Addressing the existence of complex Hadamard matrices of prime orders is a notoriously difficult problem. One of the reasons for this is that the standard construction methods developed for the study of real Hadamard matrices and various orthogonal arrays fundamentally rely on “plug-in” methods and block constructions resulting in objects of composite order.

However, a non-standard block approach was utilized by Petrescu in his PhD thesis [7], who exhibited various complex Hadamard matrices of prime orders. We recall his method as follows. Let be a positive integer, be , be an matrix, and assume that the matrix is composed of noninitial rows of a normalized complex Hadamard matrix of order . The following matrix of order

is called Petrescu’s array. The matrix is complex Hadamard if and only if the entries of and are unimodular and the following orthogonality equations are satisfied:

Let us denote by the all- matrix of order . The system of equations above can be transformed into the following equivalent, but more informative one (see [7] for details):

(1) | |||||

(2) | |||||

(3) | |||||

(4) |

Thus the lower right corner is a normal, regular matrix, and hence in view of (1) can be thought as some kind of generalized, “complex biplane”. The sum is essentially a unitary transform of , while the difference is an orthogonal matrix.

In order to prove Theorem 1.1 we take and look for some solutions of the above equations in which all of the matrix entries are some th roots of unity. First we search for candidate matrices satisfying (1) and (2), and then we search for a suitable, normalized matrix, whose six noninitial rows will constitute such that the right hand side of (3) is a matrix in which all entries are sums of some sixth roots of unity. Finally, we decompose the right hand side of (3) into some matrices and and check whether they meet the final condition (4). After implementing this rather straightforward algorithm in Mathematica, we have found the following matrix, virtually in seconds: