.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/plot_circular_ensemble.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_plot_circular_ensemble.py: Circular Ensemble ================= Defines Circular Ensemble random matrices .. GENERATED FROM PYTHON SOURCE LINES 7-13 .. code-block:: Python # Author: Alejandro Santorum Varela # License: BSD 3-Clause from skrmt.ensemble import CircularEnsemble .. GENERATED FROM PYTHON SOURCE LINES 14-16 Circular Ensemble contains random matrices introduced by Freeman Dyson as modifications of the Gaussian Ensemble. .. GENERATED FROM PYTHON SOURCE LINES 18-28 Circular Orthogonal Ensemble (COE) ---------------------------------- Random matrices of COE have real entries gaussian distributed. These matrices are invariant under orthogonal conjugation, i.e., if :math:`\mathbf{X} \in \text{COE}` and :math:`\mathbf{O}` is an orthogonal matrix, then :math:`\mathbf{O} \mathbf{X} \mathbf{O}^T` is equally distributed as :math:`\mathbf{X}`. They are also known as 1-Dyson random matrices (beta = 1). .. GENERATED FROM PYTHON SOURCE LINES 30-32 A random matrix of Circular Orthogonal Ensemble can be sampled using scikit-rmt with the following code. .. GENERATED FROM PYTHON SOURCE LINES 32-36 .. code-block:: Python coe = CircularEnsemble(beta=1, n=4) print(coe.matrix) .. rst-class:: sphx-glr-script-out .. code-block:: none [[-0.35660576-0.66292714j 0.03741206-0.41895159j -0.23281923-0.03013774j -0.16347178+0.41785612j] [ 0.03741206-0.41895159j -0.05362285-0.2116532j 0.695551 +0.05733002j 0.46645983-0.26597879j] [-0.23281923-0.03013774j 0.695551 +0.05733002j -0.36008514-0.10677441j 0.49624655-0.26549251j] [-0.16347178+0.41785612j 0.46645983-0.26597879j 0.49624655-0.26549251j -0.15016516+0.4135786j ]] .. GENERATED FROM PYTHON SOURCE LINES 37-48 Circular Unitary Ensemble (CUE) ------------------------------- Random matrices of CUE have complex entries gaussian distributed, i.e., their real part and their complex part are gaussian distributed. These matrices are invariant under unitary conjugation, i.e., if :math:`\mathbf{X} \in \text{CUE}` and :math:`\mathbf{O}` is an unitary matrix, then :math:`\mathbf{O} \mathbf{X} \mathbf{O}^T` is equally distributed as :math:`\mathbf{X}`. They are also known as 2-Dyson random matrices (beta = 2). .. GENERATED FROM PYTHON SOURCE LINES 50-52 A random matrix of Circular Unitary Ensemble can be sampled using scikit-rmt with the following code. .. GENERATED FROM PYTHON SOURCE LINES 52-56 .. code-block:: Python cue = CircularEnsemble(beta=2, n=4) print(cue.matrix) .. rst-class:: sphx-glr-script-out .. code-block:: none [[-0.22295074-0.57256518j -0.24424699+0.01609597j 0.36739606-0.38235271j -0.51258133-0.13650395j] [ 0.1273336 +0.00480154j -0.54638113+0.49540266j 0.40296706+0.18741081j 0.25150994+0.42313646j] [-0.30073519-0.3686442j -0.42216151-0.22862773j -0.32889026-0.091694j 0.62216564-0.19875148j] [ 0.12043328-0.60446737j 0.20190377+0.35337245j -0.45827167+0.44128422j -0.16101867+0.15429227j]] .. GENERATED FROM PYTHON SOURCE LINES 57-64 Circular Symplectic Ensemble (CSE) ---------------------------------- Random matrices of CSE are invariant under conjugation by the symplectic group. They are also known as 4-Dyson random matrices (beta = 4). .. GENERATED FROM PYTHON SOURCE LINES 66-68 A random matrix of Circular Symplectic Ensemble can be sampled using scikit-rmt with the following code. .. GENERATED FROM PYTHON SOURCE LINES 68-70 .. code-block:: Python cse = CircularEnsemble(beta=4, n=2) print(cse.matrix) .. rst-class:: sphx-glr-script-out .. code-block:: none [[ 1.37095220e-01-2.67012688e-01j 2.26406777e-19+0.00000000e+00j 3.32320501e-01-1.24794043e-01j -5.26180995e-01+1.74723998e-01j] [-9.71862596e-01-2.90063862e-01j -1.95225280e-01-1.42218645e-01j -9.71862596e-01-2.90063862e-01j -1.10216542e+00-2.76186562e-01j] [-6.35053489e-01+7.60061852e-01j 5.26180995e-01-1.74723998e-01j 2.71886646e-01+8.94029769e-01j 5.26180995e-01-1.74723998e-01j] [ 9.71862596e-01+2.90063862e-01j 3.32320501e-01-1.24794043e-01j -6.20374362e-18-2.77555756e-17j 6.04207147e-01+7.69235726e-01j]] .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 0.003 seconds) .. _sphx_glr_download_auto_examples_plot_circular_ensemble.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_circular_ensemble.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_circular_ensemble.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_circular_ensemble.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_