HadamardMatrix

HadamardMatrix[n]

returns an n×n Hadamard matrix.

Details and Options

  • Each entry Hrs of the Hadamard matrix is by default defined as , where , is the ^(th) bit in the binary representation of the integer , and .
  • Rows or columns of the HadamardMatrix are basis sequences of the DiscreteHadamardTransform.
  • A Hadamard matrix is its own inverse. »
  • The following options are supported:
  • MethodAutomaticspecifies the sequency ordering method
    WorkingPrecision∞the precision with which to compute matrix elements
  • Possible method settings are: "BitComplement", "Sequency".

Examples

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Basic Examples  (1)

A Hadamard matrix:

The Hadamard's basis sequences of length 128:

Options  (2)

Method  (1)

Detect sequency values of rows of a Hadamard matrix:

WorkingPrecision  (1)

By default, an exact matrix is computed:

Use machine precision:

Use arbitrary precision:

Properties & Relations  (3)

The Hadamard transform of a vector is equivalent to the vector multiplied by the Hadamard matrix:

Sylvester's construction of a non-sequency-ordered Hadamard matrix of order 4:

Hadamard matrix is its own inverse:

Wolfram Research (2012), HadamardMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/HadamardMatrix.html (updated 2014).

Text

Wolfram Research (2012), HadamardMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/HadamardMatrix.html (updated 2014).

CMS

Wolfram Language. 2012. "HadamardMatrix." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/HadamardMatrix.html.

APA

Wolfram Language. (2012). HadamardMatrix. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HadamardMatrix.html

BibTeX

@misc{reference.wolfram_2021_hadamardmatrix, author="Wolfram Research", title="{HadamardMatrix}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/HadamardMatrix.html}", note=[Accessed: 18-May-2022 ]}

BibLaTeX

@online{reference.wolfram_2021_hadamardmatrix, organization={Wolfram Research}, title={HadamardMatrix}, year={2014}, url={https://reference.wolfram.com/language/ref/HadamardMatrix.html}, note=[Accessed: 18-May-2022 ]}