Manuals/calci/HADAMARD

MATRIX("HADAMARD",order)

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H H^{T} = n I_{n}}
  

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{n}} is the n × n identity matrix and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1=\begin{bmatrix} 1 \\ \end{bmatrix}}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_2 = \begin{bmatrix} 1 & 1 \\ 1 & -1 \\ \end{bmatrix}}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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	1	1	1
1	-1	1	-1
1	1	-1	-1
1	-1	-1	1

1. 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