HadamardMatrix

HadamardMatrix[n]

returns an n×n Hadamard matrix.

Details and Options

Each entry H_rs of the Hadamard matrix is by default defined as , where , is the bit in the binary representation of the integer , and .
must be a power of two.
Rows or columns of the HadamardMatrix are basis sequences of the DiscreteHadamardTransform.
The Hadamard matrix is symmetric and orthogonal and is thus its own inverse. »
The following options are supported:
Method Automatic specify the sequency ordering method

WorkingPrecision ∞ precision at which to create entries

Examples

open allclose all

Basic Examples (1)

A Hadamard matrix:

The Hadamard's basis sequences of length 128:

Options (2)

Method (1)

Detect sequency values of rows of a Hadamard matrix:

WorkingPrecision (1)

By default, an exact matrix is computed:

Use machine precision:

Use arbitrary precision:

Properties & Relations (3)

The discrete Hadamard transform of a vector is equivalent to multiplying the vector by the Hadamard matrix:

Sylvester's construction of a Hadamard matrix of order 4:

This corresponds to the bit complement sequency ordering:

The Hadamard matrix is symmetric and orthogonal:

Because of these properties, the Hadamard matrix is its own inverse:

Top

Enable JavaScript to interact with content and submit forms on Wolfram websites. Learn how