Manuals/calci/HADAMARD

MATRIX("HADAMARD",order)

• $order$ is the order of the hadamard matrix.

Description

• This function gives the matrix satisfying the property of Hadamard.
• A Hadamard matrix is the square matrix with the entries of 1 and -1.
• Also the rows of that matrix are orthogonal.
• So H be a Hadamard matrix of order 2n.
• The transpose of H is closely related to its inverse.
• The equivalent definition for hadamard matrix is:

$HH^{T}=nI_{n}$  

where $I_{n}$ is the n × n identity matrix and $H^{T}$ is the transpose of H.

• So the possible order of the matrix is 1,2 or positive multiple of 4.
• The few examples of hadamard matrices are:
• $H_{1}={\begin{bmatrix}1\\\end{bmatrix}}$
• $H_{2}={\begin{bmatrix}1&1\\1&-1\\\end{bmatrix}}$
• $H_{3}={\begin{bmatrix}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\\\end{bmatrix}}$

Examples

• 1.MATRIX("hadamard") = 1
• 2.MATRIX("hadamard",3)

1	1	1	1
1	-1	1	-1
1	1	-1	-1
1	-1	-1	1

• 3.MATRIX("hadamard",4)

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

See Also

• ANTIDIAGONAL
• CONFERENCE
• CIRCULANT
• HANKEL

References

• Hadamard matrix