Conference Matrices – The Math


We will now look at another type of matrix known as a conference matrix.

Definition: An $n times n$ matrix $C$ is a Conference Matrix if every entry $c_{i,j}$ is either $0$, $-1$, or $1$ and $CC^T = (n-1)I_n$.

Let $C = begin{bmatrix} 0 & 1 \ 1 ^ 0 end{bmatrix}$. Then $C$ is a conference matrix as every entry of $C$ is either $0$, $-1$, or $1$ and:

(1)

begin{align} quad CC^T = begin{bmatrix} 0 & 1 \ 1 & 0 end{bmatrix} begin{bmatrix} 0 & 1\ 1 & 0 end{bmatrix} = begin{bmatrix} 1 & 0 \ 0 & 1 end{bmatrix} \ = 1I_2 end{align}

We will now begin to develop of a method for constructing conference matrices. We will first need to define a special function first.

Definition: Let $q$ be an odd prime power and let $(mathbb{Z}_q, +)$ denote the additive group of integers modulo $q$. Let $D_q$ be the set of all nonzero squares modulo $q$. The Quadratic Character Function on $mathbb{Z}_q$ is $chi_q : mathbb{Z}_q to { -1, 0, 1 }$ defined for all $x in mathbb{Z}_q$ by $chi_q(x) = left{begin{matrix} 0 & mathrm{if} : x = 0\ 1 & mathrm{if} : x in D \ -1 & mathrm{if} : x not in D end{matrix}right.$.

For example, consider the prime $q = 7$. The set of all nonzero squares modulo $q$ is:

(2)

begin{align} quad D_7 = { 1, 2, 4 } end{align}

Therefore the quadratic character function on $mathbb{Z}_7$ is:

(3)

begin{align} quad chi_7(0) &= 0 \ quad chi_7(1) &= 1 \ quad chi_(2) &= 1 \ quad chi_7(3) &= -1 \ quad chi_7(4) &= 1 \ quad chi_7(5) &= -1 \ quad chi_7(6) &= -1 \ end{align}

The following theorem gives us a method for constructing a conference matrix given a prime power $q$ of the form $q = 4n – 3$

Theorem 1: Let $q = 4n -3$ be a prime power and let $(mathbb{Z}_q, +)$ denote the additive group of integers modulo $q$. Let $infty$ denote a new point distinct from those in $mathbb{Z}_q$ and preceding the ordering of $mathbb{Z}_q$. Let $C$ be the $(q + 1) times (q + 1)$ matrix whose entries are defined by $c_{i,j} = left{begin{matrix} 1 & mathrm{if} : i = infty, j neq infty \ 1 & mathrm{if} : i neq infty, j= infty \ 0 & mathrm{if} : i = infty, j = infty \ chi_q(i-j) & mathrm{if} : i, j in mathbb{Z}_q end{matrix}right.$. Then $C$ is a conference matrix.

The condition that $q = 4n – 3$ is a prime power is equivalent to $q$ being a prime power such that $q equiv 1 pmod 4$.

For example, consider the prime $q = 5$. Clearly $q$ is a prime power and $q = 4(2) – 3$.

We aim to construct a $(q+1) times (q+1) = 6 times 6$ conference matrix. Let $D_5$ be the set of nonzero squares modulo $5$. Then:

(4)

begin{align} quad D_5 = { 1, 4 } end{align}

The quadratic character function $chi_5 : mathbb{Z}_5 to { -1, 0, 1 }$ is therefore:

(5)

begin{align} quad chi_5(0) = 0 \ quad chi_5(1) = 1 \ quad chi_5(2) = -1 \ quad chi_5(3) = -1 \ quad chi_5(4) = 1 end{align}

The matrix $C$ from Theorem 1 above will then be:

(6)

begin{align} quad C_{6 times 6} &= begin{bmatrix} 0 & 1 & 1 & 1 & 1 & 1 \ 1 & chi_5(0-0) & chi_5(0-1) & chi_5(0-2) & chi_5(0-3) & chi_5(0-4) \ 1 & chi_5(1-0) & chi_5(1-1) & chi_5(1-2) & chi_5(1-3) & chi_5(1-4) \ 1 & chi_5(2-0) & chi_5(2-1) & chi_5(2-2) & chi_5(2-3) & chi_5(2-4) \ 1 & chi_5(3-0) & chi_5(3-1) & chi_5(3-2) & chi_5(3-3) & chi_5(3-4) \ 1 & chi_5(4-0) & chi_5(4-1) & chi_5(4-2) & chi_5(4-3) & chi_5(4-4) \ end{bmatrix} \ &= begin{bmatrix} 0 & 1 & 1 & 1 & 1 & 1 \ 1 & chi_5(0) & chi_5(4) & chi_5(3) & chi_5(2) & chi_5(1) \ 1 & chi_5(1) & chi_5(0) & chi_5(4) & chi_5(3) & chi_5(2) \ 1 & chi_5(2) & chi_5(1) & chi_5(0) & chi_5(4) & chi_5(3) \ 1 & chi_5(3) & chi_5(2) & chi_5(1) & chi_5(0) & chi_5(4) \ 1 & chi_5(4) & chi_5(3) & chi_5(2) & chi_5(1) & chi_5(0) \ end{bmatrix} \ &= begin{bmatrix} 0 & 1 & 1 & 1 & 1 & 1 \ 1 & 0 & 1 & -1 & -1 & 1 \ 1 & 1 & 0 & 1 & -1 & -1 \ 1 & -1 & 1 & 0 & 1 & -1 \ 1 & -1 & -1 & 1 & 0 & 1 \ 1 & 1 & -1 & -1 & 1 & 0 \ end{bmatrix} end{align}



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