`hadamard`

Hadamard matrix of +1 and -1 entries.

Category: Mathematics > Elementary Math

Syntax

H = hadamard(n)

Returns an n-by-n matrix H whose entries are 1 or -1 and whose rows are mutually orthogonal.

Description

The defining property is that distinct rows are orthogonal, so H*H' is n times the identity. Such matrices are the basis of the Walsh-Hadamard transform and of error-correcting codes and experimental designs.

For powers of two the matrix is built by the doubling rule [H H; H -H]. Other orders use known constructions, and an order for which no matrix of this type is available is an error.

Input arguments

n: Order of the matrix. It must be 1, 2, or a multiple of 4 of the form 2^e, 12*2^e or 20*2^e.

Output arguments

H: Matrix satisfying H'*H == n*eye(n).

Example

Order 4 and its orthogonality

H = hadamard(4)
H' * H

Output

H = 4×4

     1     1     1     1
     1    -1     1    -1
     1     1    -1    -1
     1    -1    -1     1
ans = 4×4

     4     0     0     0
     0     4     0     0
     0     0     4     0
     0     0     0     4

See also

hilb magic gallery kron