# Stochastic Matrices

## Stochastic Matrices

RandomMatrix.randStochasticFunction
randStochastic(n; type, norm)
• n: dimension
• type : default type = 3, 3 for doubly randStochastic, 1 for row, 2 for column stochastic
• norm : default false, if set to true, the matrix will be normalized by $\sqrt{n}$ (not a typo)

Examples

Generates a 3 by 3 random doubly stochastic matrix

randStochastic(3)

3×3 Matrix{Float64}:
0.132593  0.216041  0.651367
0.484097  0.320777  0.195126
0.261495  0.537825  0.20068

Generates a 3 by 3 normalized random row stochastic matrix

randStochastic(3,type = 1)

3×3 Matrix{Float64}:
0.220849   0.146942  0.632209
0.188052   0.26294   0.549008
0.0170714  0.524574  0.458355

Generates a 3 by 3 normalized random column stochastic matrix

randStochastic(3,type=2,norm=true)

3×3 Matrix{Float64}:
0.583396  0.608739  0.732921
0.672821  0.078786  0.302657
0.475834  1.04453   0.696473
source

## RMT: Circular Law for Doubly Stochastic Random Matrices

Let $X$ be a matrix sampled uniformly from the set of doubly stochastic matrices of size $n \times n$. The empirical spectral distribution of the normalized matrix $\sqrt{n}(X-\mathbf{E} X)$ converges almost surely to the circular law.

For reference, see the paper by Hoi H. Nguyen Random doubly stochastic matrices: The circular law