## Quaternary complex Hadamard matrices

by Pekka Lampio, Ferenc Szöllösi and Patric Östergård.

This Wiki space is for publishing results that do not fit in our paper "Quaternary complex Hadamard matrices of orders 10, 12, and 14".

**Contents**

### Definitions

A **complex Hadamard matrix**is any complex
\( n \times n \)
matrix
\( H \)
satisfying two conditions:

unimodularity (the modulus of each entry is unity): \( \|H_{jk}\| = 1 \) for \( j,k=1,2,\ldots,n \)

orthogonality: \( HH^* = n I \) ,

where \( H^* \) denotes the Hermitian transpose of H and \( I \) is the identity matrix. This concept is a generalization of the Hadamard matrix.

A **Butson-type Hadamard matrix**is a complex Hadamard matrix where the matrix entries are q:th roots of unity, that is, the entries are solutions to the equation
\( q^n = 1 \)
. An
\( n \times n \)
Butson-type Hadamard matrix over q:th roots of unity is denoted by BH(q,n).

Two complex Hadamard matrices are called **equivalent**, written
\( H_1 \cong H_2 \)
, if there exist diagonal unitary matrices
\( D_1, D_2 \)
and permutation matrices
\( P_1, P_2 \)
such that
\( H_1 = D_1 P_1 H_2 P_2 D_2 \)
.

Any complex Hadamard matrix is equivalent to a **dephased** matrix, in which all elements in the first row and first column are equal to unity.

Two complex Hadamard matrices
\( H \)
and
\( K \)
are called **ACT-equivalent**, if
\( H \)
is equivalent to any of the following:
\( K \)
; or the **A**djoint of
\( K \)
; or the **C**onjugate of
\( K \)
; or the **T**ranspose of
\( K \)
.

### Classification

The table below summarizes the classification of Butson-type Hadamard matrices over fourth roots of unity for orders up to 14.

### References

[1] P. H. J. Lampio, F. Szöllősi, P. R. J. Östergård: *The quaternary complex Hadamard matrices of orders 10, 12, and 14.* Discrete Math., **313**:189–206 (2013).

[2] P. H. J. Lampio: *Classification of difference matrices and complex Hadamard matrices.* PhD thesis, Aalto University (2015).

[3] F. Szöllősi: *On quaternary complex Hadamard matrices of small orders*, Adv. Math. Commun., **5**,309-315 (2011).