Hadamard Matrix — Definition, Formula & Examples

A Hadamard matrix is a square matrix whose every entry is either

+1

-1

and whose rows are all mutually orthogonal. When you multiply it by its own transpose, you get

n

times the identity matrix, where

n

is the matrix size.

An $n \times n$ matrix $H$ with entries $h_{ij} \in \{+1, -1\}$ is a Hadamard matrix if it satisfies $HH^T = nI_n$ , where $I_n$ is the $n \times n$ identity matrix and $H^T$ denotes the transpose of $H$ .

Key Formula

HH^T = nI_n

Where:

$H$ = An n × n matrix with all entries +1 or −1
$H^T$ = The transpose of H
$n$ = The order (size) of the matrix
$I_n$ = The n × n identity matrix

How It Works

To verify that a matrix is Hadamard, check two things: every entry must be

+1

-1

, and the dot product of any two distinct rows must equal zero. The dot product of any row with itself equals

n

, so the product

HH^T

yields

nI_n

. Hadamard matrices exist only for

n = 1

n = 2

, or

n

a multiple of

4

. The famous Hadamard conjecture asserts that a Hadamard matrix exists for every positive multiple of

4

, but this remains unproven.

Worked Example

Problem: Verify that the following 2×2 matrix is a Hadamard matrix:

H = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}

Check entries: Every entry of H is either +1 or −1, so the entry condition is satisfied.

Compute H times H transpose: Since H is symmetric here,

H^T = H

. Compute the product:

HH^T = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}

Compare with nI: Here

n = 2

, and

2I_2 = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}

. The result matches, confirming the orthogonality condition.

HH^T = 2I_2 \checkmark

Answer: H is a Hadamard matrix of order 2 because all entries are ±1 and

HH^T = 2I_2

Why It Matters

Hadamard matrices appear in error-correcting codes (like Reed–Muller codes), signal processing, and experimental design in statistics. In quantum computing, the

2 \times 2

Hadamard matrix (scaled by

\frac{1}{\sqrt{2}}

) is one of the most fundamental quantum gates, creating superposition states from basis states.

Common Mistakes

Mistake: Assuming a Hadamard matrix can exist for any size n.

Correction: Hadamard matrices can only exist when n = 1, n = 2, or n is a multiple of 4. There is no 3×3 or 5×5 Hadamard matrix.