Under construction

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

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

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        

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