Hadamard Matrix

Generate Hadamard matrix of a certain size.

hadamard(n)

Arguments

n

An integer of the form 2^e, 12*2^e, or 20*2^e

Details

An n-by-n Hadamard matrix with n>2 exists only if rem(n,4)=0. This function handles only the cases where n, n/12, or n/20 is a power of 2.

Value

Matrix of size n-by-n of orthogonal columns consisting of 1 and -1 only.

Note

Hadamard matrices have applications in combinatorics, signal processing, and numerical analysis.

See also

hankel, Toeplitz

Examples

hadamard(4)
#>      [,1] [,2] [,3] [,4]
#> [1,]    1    1    1    1
#> [2,]    1   -1    1   -1
#> [3,]    1    1   -1   -1
#> [4,]    1   -1   -1    1
H <- hadamard(8)
t(H) 
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
#> [1,]    1    1    1    1    1    1    1    1
#> [2,]    1   -1    1   -1    1   -1    1   -1
#> [3,]    1    1   -1   -1    1    1   -1   -1
#> [4,]    1   -1   -1    1    1   -1   -1    1
#> [5,]    1    1    1    1   -1   -1   -1   -1
#> [6,]    1   -1    1   -1   -1    1   -1    1
#> [7,]    1    1   -1   -1   -1   -1    1    1
#> [8,]    1   -1   -1    1   -1    1    1   -1