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


  • 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 TemplateBox[{x}, Transpose].J.x=J, 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.

Background & Context


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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,


Wolfram Research (2015), CircularQuaternionMatrixDistribution, Wolfram Language function,


Wolfram Language. 2015. "CircularQuaternionMatrixDistribution." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2015). CircularQuaternionMatrixDistribution. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_circularquaternionmatrixdistribution, author="Wolfram Research", title="{CircularQuaternionMatrixDistribution}", year="2015", howpublished="\url{}", note=[Accessed: 23-May-2024 ]}


@online{reference.wolfram_2024_circularquaternionmatrixdistribution, organization={Wolfram Research}, title={CircularQuaternionMatrixDistribution}, year={2015}, url={}, note=[Accessed: 23-May-2024 ]}