Matrix-valued Distributions
GUE
RandomMatrix.GUE
— TypeGUE <: ContinuousMatrixDistribution
GUE(n)
n
: dimension- The Gaussian Unitary Ensemble (
GUE
) is an ensemble of random $n \times n$ Hermitian matrices $M_{n}$ in which the upper-triangular entries are iid with distribution $N(0,1)_{\mathbf{C}}$, and the diagonal entries are iid with distribution $N(0,1)_{\mathbf{R}}$, and independent of the upper-triangular ones
rand(M::GUE, norm::bool)
norm
: defaultfalse
, ifnorm
set totrue
, then the matrix will be normlaized with $\operatorname{min}(n,m)^{-1/2}$.
Examples
Generate a 3 by 3 random matrix from GUE(3)
rand(GUE(3))
3×3 Hermitian{ComplexF64, Matrix{ComplexF64}}:
-0.883413+0.0im 1.09872+0.874884im -0.1985-1.04778im
1.09872-0.874884im 1.55483+0.0im -0.488532+1.18694im
-0.1985+1.04778im -0.488532-1.18694im -0.0823873+0.0im
rand(GUE(2),norm=true)
2×2 Hermitian{ComplexF64, Matrix{ComplexF64}}:
-0.457089+0.0im -0.672713-0.102234im
-0.672713+0.102234im 0.380126+0.0im
GOE
RandomMatrix.GOE
— TypeGOE <: ContinuousMatrixDistribution
GOE(n)
n
: dimension- The Gaussian Orthogonal Ensemble (
GOE
) is an ensemble of random $n \times n$ Symmetric matrices $M_{n}$ in which the upper-triangular entries are iid with distribution $N(0,1)_{\mathbf{R}}$, and the diagonal entries are iid with distribution $N(0,2)_{\mathbf{R}}$, and independent of the upper-triangular ones
rand(M::GOE, norm::bool)
norm
: defaultfalse
, ifnorm
set totrue
, then the matrix will be normlaized with $\operatorname{min}(n,m)^{-1/2}$.
Examples
Generate a 3 by 3 random matrix from GOE(3)
rand(GOE(3))
3×3 Symmetric{Float64, Matrix{Float64}}:
-1.62391 -0.451433 0.863883
-0.451433 0.0271799 -0.524854
0.863883 -0.524854 -0.00930624
rand(GOE(3),norm=true)
3×3 Symmetric{Float64, Matrix{Float64}}:
0.302141 0.152634 -0.711679
0.152634 -0.0629327 0.103075
-0.711679 0.103075 1.51861
Haar
RandomMatrix.Haar
— TypeHaar(beta,n)
- Uniform distribution on O(n) (
beta = 1
), U(n) (beta = 2
) beta
: 1 for Orthogonal, 2 for Unitaryn
: dimension
# Examples
# Generate a 100 by 100 random Unitary Matrix uniformly from U(n)
rand(Haar(2,100))
# Generate a 100 by 100 random Orthogonal Matrix uniformly from O(n)
rand(Haar(1,100))
COE
RandomMatrix.COE
— FunctionCOE
n
: Dimension- Equivalent to
Haar
(1,n)
CUE
RandomMatrix.CUE
— FunctionCUE
n
: Dimension- Equivalent to
Haar
(2,n)