Language Reference

HADAMARD Function

returns a Hadamard matrix

HADAMARD( n, <, i>)

The inputs to the HADAMARD function are as follows:

n

specifies the order of the Hadamard matrix. Specify such that or a multiple of 4 and any of the following hold:

• 
• is prime
• is prime and
• , where is any above

When any other is specified, the HADAMARD function returns a zero.

i

specifies the row number to return. When is not specified or is negative, the full matrix is returned.

The HADAMARD function returns a Hadamard matrix, which is an matrix consisting entirely of the values 1 and - 1. The columns of a Hadamard matrix are all orthogonal. Hadamard matrices are frequently used to make orthogonal array experimental designs for two-level factors. For example, the following IML statements create a Hadamard matrix:

  
    h = hadamard(12); 
    print h[format=2.];
 

The results are as follows:

  
                                        H 
  
                        1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 
                        1  1 -1  1 -1 -1 -1  1  1  1 -1  1 
                        1  1  1 -1  1 -1 -1 -1  1  1  1 -1 
                        1 -1  1  1 -1  1 -1 -1 -1  1  1  1 
                        1  1 -1  1  1 -1  1 -1 -1 -1  1  1 
                        1  1  1 -1  1  1 -1  1 -1 -1 -1  1 
                        1  1  1  1 -1  1  1 -1  1 -1 -1 -1 
                        1 -1  1  1  1 -1  1  1 -1  1 -1 -1 
                        1 -1 -1  1  1  1 -1  1  1 -1  1 -1 
                        1 -1 -1 -1  1  1  1 -1  1  1 -1  1 
                        1  1 -1 -1 -1  1  1  1 -1  1  1 -1 
                        1 -1  1 -1 -1 -1  1  1  1 -1  1  1
 

The first column is an intercept and the next 11 columns form an orthogonal array experimental design for 11 two-level factors in 12 runs, .

To request the 17th row of a Hadamard matrix of order 448, use the following statement:

  
    h = hadamard(448, 17);