freealg.distributions.KestenMcKay.matrix

KestenMcKay.matrix(size, seed=None)

Generate matrix with the spectral density of the distribution.

Parameters:

sizeint

Size \(n\) of the matrix.

seedint, default=None

Seed for random number generator.

Returns:

Anumpy.ndarray

A matrix of the size \(n \times n\).

Notes

If the parameter \(d\) is even, the matrtix is generated from

\[\mathbf{A} = \sum_{i=1}^{d/2} \mathbf{O}_i + \mathbf{O}_o^{\intercal},\]

where \(\mathbf{O}_i\) are randomly generated orthogonal matrices with Haar. This method is fast but \(d\) has to be even.

If all other \(d\), the following is used:

\[\mathbf{A} = \mathbf{P} \mathbf{O} \mathbf{D} \mathbf{O}^{-1} \mathbf{P},\]

where \(\mathbf{D}\) is diagonal matrix with entries \(\pm 1\), \(\mathbf{O}\) is orthogonal with Haar measure, and \(\mathbf{P}\) is a projection matrix. For more details, see Section 5 and 6 of [1].

The orthogonal matrices are genrated using the method of [2].

References

[1]

Iris S. A. Longoria and James A. Mingo, Freely Independent Coin Tosses, Standard Young Tableaux, and the Kesten–McKay Law. https://arxiv.org/abs/2009.11950

[2]

Francesco Mezzadri. How to generate random matrices from the classical compact groups. https://arxiv.org/abs/math-ph/0609050

Examples