# CircularQuaternionMatrixDistribution

CircularQuaternionMatrixDistribution[n]

represents a circular quaternion matrix distribution with matrix dimensions {2 n,2 n} over the field of complex numbers.

# Details

• CircularQuaternionMatrixDistribution is also known as circular quaternion ensemble, or CQE.
• CircularQuaternionMatrixDistribution represents a uniform distribution over the compact symplectic square matrices of dimension n. It is also known as the Haar measure on the unitary symplectic group .
• Each realization of CircularQuaternionMatrixDistribution is represented as a unitary matrix that preserves the symplectic form , where is a symplectic matrix KroneckerProduct[{{0,-1},{1,0}},IdentityMatrix[n]].
• The dimension parameter n can be any positive integer.
• CircularQuaternionMatrixDistribution can be used with such functions as MatrixPropertyDistribution and RandomVariate.

# Examples

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## Basic Examples(2)

Generate a pseudorandom CQE matrix:

It is unitary and preserves the symplectic matrix :

Represent the eigenvalues of a random matrix by MatrixPropertyDistribution and sample from it:

## Scope(3)

Generate a random matrix from unitary symplectic group :

Generate a set of random matrices from unitary symplectic group:

Compute statistical properties numerically:

## Properties & Relations(2)

Distribution of phase angle of the eigenvalues:

Compute the spacing between eigenvalues:

Compare the histogram of sample level spacings with the closed form, also known as Wigner surmise for Dyson index 2:

For eigenvectors of CircularQuaternionMatrixDistribution with dimension large, the scaled modulus of the elements is distributed:

Compare the histogram with PDF of ChiSquareDistribution:

Wolfram Research (2015), CircularQuaternionMatrixDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/CircularQuaternionMatrixDistribution.html.

#### Text

Wolfram Research (2015), CircularQuaternionMatrixDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/CircularQuaternionMatrixDistribution.html.

#### CMS

Wolfram Language. 2015. "CircularQuaternionMatrixDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CircularQuaternionMatrixDistribution.html.

#### APA

Wolfram Language. (2015). CircularQuaternionMatrixDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CircularQuaternionMatrixDistribution.html

#### BibTeX

@misc{reference.wolfram_2024_circularquaternionmatrixdistribution, author="Wolfram Research", title="{CircularQuaternionMatrixDistribution}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/CircularQuaternionMatrixDistribution.html}", note=[Accessed: 18-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_circularquaternionmatrixdistribution, organization={Wolfram Research}, title={CircularQuaternionMatrixDistribution}, year={2015}, url={https://reference.wolfram.com/language/ref/CircularQuaternionMatrixDistribution.html}, note=[Accessed: 18-May-2024 ]}