.. 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 [[-0.12645976+0.41434039j 0.07087014+0.3325673j 0.25825756-0.40693271j -0.27854225+0.62195611j] [ 0.07087014+0.3325673j 0.04292674-0.2518253j 0.75512459+0.2274324j 0.4418379 -0.04425334j] [ 0.25825756-0.40693271j 0.75512459+0.2274324j 0.29967872+0.03800235j -0.21876984-0.0815991j ] [-0.27854225+0.62195611j 0.4418379 -0.04425334j -0.21876984-0.0815991j -0.1275336 -0.51732219j]] .. 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 [[-0.36058796-0.43195498j -0.13631854+0.2362593j -0.23617952-0.70568448j 0.21376232-0.09759275j] [-0.20823174+0.45314635j -0.02050427-0.63267576j 0.16983497-0.4723054j -0.22228646-0.22197142j] [ 0.53267569-0.33648245j 0.35472113-0.39610975j -0.36727905-0.04732658j 0.13519559-0.40607217j] [-0.02975125-0.19193797j 0.4713975 -0.14125912j -0.1582397 -0.17927307j -0.43322638+0.6893774j ]] .. 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 [[-1.32223241e-01-3.93257553e-01j -7.64256279e-18-1.99493200e-17j -5.10882402e-01-7.51682255e-02j -3.09206758e-03-3.90820188e-01j] [ 6.25460078e-01+3.89337197e-02j 3.78659161e-01-3.18089327e-01j 6.25460078e-01+3.89337197e-02j -2.80852595e-01-8.12799161e-01j] [ 8.84844130e-01-1.72706143e-01j 3.09206758e-03+3.90820188e-01j 1.54435589e+00+3.22003691e-01j 3.09206758e-03+3.90820188e-01j] [-6.25460078e-01-3.89337197e-02j -5.10882402e-01-7.51682255e-02j -1.97020323e-17+0.00000000e+00j 1.03347348e+00+2.46835466e-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 `_