Manova Ensemble

Defines Manova Ensemble random matrices

# Author: Alejandro Santorum Varela
# License: BSD 3-Clause

from skrmt.ensemble import ManovaEnsemble

Manova Ensemble random matrices are considered to be double Wishart random matrices.

Manova Real Ensemble (MRE)

Random matrices of MRE are formed by sampling two random real standard guassian matrices (\(\mathbf{X}\) and \(\mathbf{Y}\)) of size \(m \times n_1\) and \(m \times n_2\) respectively. Then, matrix \(\mathbf{A} = \dfrac{\mathbf{X}\mathbf{X}^T}{\mathbf{X}\mathbf{X}^T + \mathbf{Y}\mathbf{Y}^T}\) generates a matrix of the MRE.

They are also known as 1-Jacobi random matrices (beta = 1).

A random matrix of Manova Real Ensemble can be sampled using scikit-rmt with the following code.

mre = ManovaEnsemble(beta=1, m=3, n1=5, n2=5)
print(mre.matrix)

[[ 0.37357168 -0.0229562  -0.12886666]
 [ 0.10982747  0.65232374  0.22545803]
 [ 0.10811206  0.11454821  0.78460693]]

Manova Complex Ensemble (MCE)

Random matrices of MCE are formed by sampling two random complex standard guassian matrices (\(\mathbf{X}\) and \(\mathbf{Y}\)) of size \(m \times n_1\) and \(m \times n_2\) respectively. Then, matrix \(\mathbf{A} = \dfrac{\mathbf{X}\mathbf{X}^T}{\mathbf{X}\mathbf{X}^T + \mathbf{Y}\mathbf{Y}^T}\) generates a matrix of the MCE.

They are also known as 2-Jacobi random matrices (beta = 2).

A random matrix of Manova Complex Ensemble can be sampled using scikit-rmt with the following code.

mce = ManovaEnsemble(beta=2, m=3, n1=5, n2=5)
print(mce.matrix)

[[ 0.42539464+0.1414285j   0.14444995-0.05260573j -0.26449982+0.3652106j ]
 [ 0.0397875 +0.22067762j  0.39708865-0.10969292j -0.19966326+0.60321587j]
 [ 0.04435281-0.0831098j  -0.14731224-0.13789568j  0.73233881-0.03173559j]]

Manova Quaternion Ensemble (MQE)

Random matrices of MQE are formed by sampling two random complex standard guassian matrices (\(\mathbf{X_1}\) and \(\mathbf{X_2}\)), both of size \(m \times n_1\). Another two random complex standard guassian matrices (\(\mathbf{Y_1}\) and \(\mathbf{Y_2}\)), both of size \(m \times n_2\), are sampled. They are stacked forming matrices \(\mathbf{X}\) and \(\mathbf{Y}\):

\(\mathbf{X} = (\mathbf{X_1}\ \mathbf{X_2}; -\mathbf{X_2}^*\ \mathbf{X_1}^*)\)

\(\mathbf{Y} = (\mathbf{Y_1}\ \mathbf{Y_2}; -\mathbf{Y_2}^*\ \mathbf{Y_1}^*)\)

Finally, matrix \(\mathbf{A} = \dfrac{\mathbf{X}\mathbf{X}^T}{\mathbf{X}\mathbf{X}^T + \mathbf{Y}\mathbf{Y}^T}\) generates a matrix of the MQE.

They are also known as 4-Jacobi random matrices (beta = 4).

A random matrix of Manova Quaternion Ensemble can be sampled using scikit-rmt with the following code.

mqe = ManovaEnsemble(beta=4, m=2, n1=5, n2=5)
print(mqe.matrix)

[[ 0.51844198+0.01160693j -0.01631465+0.00406845j -0.00299089-0.00824204j
   0.05202532+0.05268844j]
 [-0.02577827-0.00708927j  0.50089801+0.01283598j -0.08565466-0.09086775j
   0.00342764+0.00592337j]
 [ 0.00299089-0.00824204j -0.05202532+0.05268844j  0.51844198-0.01160693j
  -0.01631465-0.00406845j]
 [ 0.08565466-0.09086775j -0.00342764+0.00592337j -0.02577827+0.00708927j
   0.50089801-0.01283598j]]