# Under construction

## Hermitian

RandomMatrix.randHermitianFunction
randHermitian(d, n; diag, norm )

randHermitian(n; norm)
• d : entry distribution
• n : dimensions
• norm : default false, if norm set to true, then the matrix will be normalized with $n^{-1/2}$.
• diag : default diag = d, diagonal entry distribution. To use a different distribution (say Circular(2)) for digonal elements, set diag = Circular(2). The diagonal entries will always be forced to have imgainary part 0.
• See also GUE

Examples

Generates a 2 by 2 random Hermitian matrix with off-diagonal entries from the Standard Complex Gaussian, and Standard Normal on the diagonal.

randHermitian(2)

2×2 Hermitian{ComplexF64, Matrix{ComplexF64}}:
0.382095+0.0im        -0.708469-0.0636734im
-0.708469+0.0636734im   0.336952+0.0im

Generate a 3 by 3 Hermitian matrix, with off-diagonal entries Circular(1) and diagonal entries uniformly -1 or 1.

randHermitian(Circular(1),3,diag = (-1,1))

3×3 Hermitian{ComplexF64, Matrix{ComplexF64}}:
1.0+0.0im       1.56259-0.676099im  1.39468-0.295073im
1.56259+0.676099im     -1.0+0.0im       1.53369+0.296583im
1.39468+0.295073im  1.53369-0.296583im     -1.0+0.0im

Generate a random 2 by 2 Symmetric Matrix with entries Poisson(2) rvs. This can also be done with randSymmetric(Poisson(2),3)

randHermitian(Poisson(2),3)

3×3 Hermitian{Int64, Matrix{Int64}}:
3  1  0
1  1  2
0  2  1

Entries uniformly from $\{1,2,3,...,10\}$

randHermitian(1:10,2)

2×2 Hermitian{Int64, Matrix{Int64}}:
10  7
7  6

Entries either -1 or pi with equal probability

randHermitian([-1,pi],3)

3×3 Hermitian{Float64, Matrix{Float64}}:
3.14159  -1.0      -1.0
-1.0      -1.0       3.14159
-1.0       3.14159   3.14159
source

## Symmetric

RandomMatrix.randSymmetricFunction
randSymmetric(d, n; Diag, norm)

randSymmetric(n; norm)
• d : entry distribution
• n : dimensions
• norm : default false, if norm set to true, then the matrix will be normalized with $n^{-1/2}$.
• diag : default diag = d, the distribution for diagonal entries. To use a different distribution (say Binomial) for digonal elements, set diag = Binomial(1,0.5)
• See also GOE

Examples

Generates a 3 by 3 random Symmetric matrix with entries from the Standard Gaussian.

randSymmetric(3)

3×3 Symmetric{Float64, Matrix{Float64}}:
-0.230698  -1.72846     0.306362
-1.72846    0.0845915  -0.0116108
0.306362  -0.0116108  -0.559046
source

## Elliptic Matrices

RandomMatrix.randEllipticFunction
randElliptic(d, n; r , diag, norm)

randElliptic(n; r, norm)
• d : default Normal(), entry distribution
• n : dimensions
• r : default 0.5, the correlation of $H_{ij},H_{ji}$ pairs
• norm : default false, if norm set to true, then the matrix will be normalized with $n^{-1/2}$.
• diag : default diag=d, the distribution for diagonal entries.

Examples

Generate a random elliptic matrix, with entries from $\mathscr{N}(0,1)$ and $\rho(H_{ij},H_{ji})=0.5$

randElliptic(500)

500×500 Matrix{Float64}:
2.03417    -0.424289    1.28267   …  -0.114754  -1.96059
0.44479    -1.45563     1.32828       1.00149    0.45786
1.56525     0.0832211  -0.186738     -1.3914     1.04151
-0.11633    -0.483301   -1.81348      -1.57536    0.514818
⋮                                 ⋱
1.24274    -0.411623   -1.04984      -0.812778  -1.84479
-0.0817287  -0.254886    0.674914      0.756269  -0.0296209
-1.48281     0.51675    -1.58041       0.156923   0.244599
0.852339    1.04593    -0.119082      1.43634    0.114493

Generate a normalized random elliptic matrix, with entries Poisson(10) and $\rho(H_{ij},H_{ji})=0.1$

using Distributions
randElliptic(Poisson(10),500, r=0.1 , norm=true)

500×500 Matrix{Float64}:
0.268328    -0.0413153  -0.0175096   …   0.0190835   0.0201304
-0.00599949   0.447214   -0.0175878       0.0112805   0.100704
-0.0258879    0.0219927   0.402492       -0.04749    -0.050853
0.0219071    0.0119609  -0.00448502      0.0043233   0.0404757
⋮                                    ⋱
-0.0145467    0.0800297   0.00247891      0.0189267   0.071565
0.016412     0.0334019   0.0663348      -0.0180889   0.023773
-0.0485914   -0.0575288  -0.0409827       0.491935   -0.0969691
0.0405447    0.0503843   0.00624668      0.0558304   0.402492

source

(This is a old typo) See randElliptic for more details.

source

## RMT: Semicircle Law

The semi-circular law, lecture notes by Terence Tao

## RMT : Elliptic Law

THE ELLIPTIC LAW by NGUYEN and O’ROURKE