Equivalent and Normalized Hadamard Matrices

Recall from the Hadamard Matrices page that a Hadamard matrix of order $n$ is an $n \times n$ matrix $H$ with the properties that $h_{ij} = \pm 1$ for all $i, j \in \{ 1, 2, ..., n \}$ and $HH^T = nI_n$.

We stated two important results on the page above:

• Interchanging any two rows or any two columns of a Hadamard matrix yields another Hadamard matrix.

• Multiplying any row or any column of a Hadamard matrix by $-1$ yields another Hadamard matrix.

With these results, we can categorize equivalence of Hadamard matrices.

Definition: Two $n \times n$ Hadamard matrices $H$ and $H'$ are Equivalent if $H'$ can be obtained by $H$ by interchanging rows/columns of $H$ or multiplying rows/columns of $H$ by $-1$.

Recall the following Hadamard matrix:

(1)

\begin{align} \quad H = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \end{align}

Some examples of Hadamard matrices equivalent to the one above are:

(2)

\begin{align} \quad \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix} \quad , \quad \begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix} \quad , \quad \begin{bmatrix} -1 & -1 \\ 1 & -1 \end{bmatrix} \quad , \quad \begin{bmatrix} -1 & 1 \\ -1 & -1 \end{bmatrix} \end{align}

From the results stated at the top of the page, it is completely possible to take any Hadamard matrix and obtain an equivalent Hadamard matrix whose first row consists entirely of positive $1$s. Such Hadamard matrices are defined below.

Definition: A Hadamard matrix is said to be Normalized if the first row consists entirely of positive $1$s.

If we consider the Hadamard matrix $\begin{bmatrix} 1 & -1 \\1 & 1 \end{bmatrix}$ from above we can see that the corresponding normalized equivalent Hadamatrx matrix would be:

(3)

\begin{align} \quad \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \end{align}