under construction

randUnitary(n::Int)

• Generates a n by n random Unitary matrix
• Equivalent to rand(Haar(2,n)), see Haar and CUE
• For orthogonal matrices, use randOrthogonal or rand(Haar(1,n)) instead
• The algorithm is disccused in How to Generate Random Matrices from the Classical Compact Groups

Examples

Generate a 3 by 3 random Unitary matrix

randUnitary(3)

3×3 Matrix{ComplexF64}:
 -0.149398+0.0572715im  -0.0935861+0.629201im  -0.257255-0.709625im
  0.337035-0.342606im     -0.36366+0.599236im  -0.100838+0.517231im
  -0.17097+0.845103im   -0.0767105+0.313259im   0.247081+0.3025im

randOrthogonal(n::Int)

• Generates a n by n random Orthogonal matrix
• Equivalent to rand(Haar(1,n)), see Haar and COE
• For unitary matrices, use randUnitary or rand(Haar(2,n)) instead
• The algorithm is disccused in How to Generate Random Matrices from the Classical Compact Groups

Examples

Generates a 3 by 3 random Orthogonal matrix

randOrthogonal(3)

3×3 Matrix{Float64}:
 -0.875553  0.112448   0.469853
 -0.147915  0.863441  -0.482277
  0.459921  0.491757   0.739356

randPermutation(n; fix) 

• n : dimension
• fix : a keyword argument, default is set to fix = 0. If fix = x, randPermutation(n,x) will have atleast x fixed points

Examples

Generates a random 5 by 5 permutation matrix

randPermutation(5)

5×5 SparseArrays.SparseMatrixCSC{Int8, Int64} with 5 stored entries:
 ⋅  ⋅  ⋅  ⋅  1
 ⋅  1  ⋅  ⋅  ⋅
 1  ⋅  ⋅  ⋅  ⋅
 ⋅  ⋅  1  ⋅  ⋅
 ⋅  ⋅  ⋅  1  ⋅

Generates Generates a random 10 by 10 permutation matrix with atleast 7 fix points

randPermutation(10, fix = 7)

10×10 SparseArrays.SparseMatrixCSC{Int8, Int64} with 10 stored entries: 
 1  ⋅  ⋅  ⋅  ⋅  ⋅  ⋅  ⋅  ⋅  ⋅
 ⋅  1  ⋅  ⋅  ⋅  ⋅  ⋅  ⋅  ⋅  ⋅
 ⋅  ⋅  ⋅  ⋅  ⋅  ⋅  ⋅  ⋅  1  ⋅
 ⋅  ⋅  ⋅  1  ⋅  ⋅  ⋅  ⋅  ⋅  ⋅
 ⋮              ⋮
 ⋅  ⋅  ⋅  ⋅  ⋅  ⋅  1  ⋅  ⋅  ⋅
 ⋅  ⋅  ⋅  ⋅  ⋅  ⋅  ⋅  1  ⋅  ⋅
 ⋅  ⋅  1  ⋅  ⋅  ⋅  ⋅  ⋅  ⋅  ⋅
 ⋅  ⋅  ⋅  ⋅  ⋅  ⋅  ⋅  ⋅  ⋅  1