Few days back I was playing with the eigshow demo in Matlab. It is a pretty useful demonstration of eigenvalue and singular value problems for 2 by 2 matrices. A good description of the eigshow demo in Matlab is here. I was a little surprised not to find an “eigshow” implementation in my favorite computing language — Python. So I thought it might be a good idea to write my own, and here is the result.
The figure below shows the basic structure of the program which is somewhat similar to Matlab’s eigshow. There are some differences too, such as it displays the eigenvalues and eigenvectors in the eigen mode, and the singular values and singular vectors in the SVD mode. The program also displays the rank, trace, and determinant along with the matrix. It also gives a visible warning if the eigen/singular values and/or vectors are complex (as it can only plots the real values and vectors).
(Please continue reading to know about basic function and code)
For the GUI, I decided to enforce a TkAgg backend even though I used matplotlib for plotting. As it is not possible to implement a drop-down menu using the matplotlib widgets (as of this writing), I have used Tkinter widgets for the drop-down menu. The drop-down menu is used for selecting a matrix from a variety of matrices). Also, since I ended up using Tkinter, I decided to render the matplotlib figure objects on Tkinter’s canvas. Partly, the reason for using Tkinter is also because it is the standard GUI toolkit distributed with Python. Also, in order to rotate the vector x, I resorted to using mouse-click events as I wasn’t sure how to implement mouse-drag with continuous sampling of locations (if anyone happens to know how to do it, your inputs would be of great help).
One can execute the code from any python shell (such as the IPython shell or qtconsole) or an IDE. Once it is in the namespace, one can also pass any other 2×2 matrix by calling eigshow(A), where A is any 2×2 Numpy array/matrix.
Finally, here is the code for eigshow in Python. Granted that it is not highly efficient (it was never meant to be), and I could have used OOD to write a much cleaner code, it is a working code. Please feel free to download it and modify it as you may please. Also, suggestions are always welcome (thank you in advance).
[Update: While posting the code in the post, the indentation got messed up at several places. Unfortunately, for Python codes that will be a problem. So, if you are interested, please download the code from here.]
"""
Created on Wed Jan 16 17:40:27 2013
Author: Indranil Sinharoy
Licence: BSD
"""
from __future__ import division
#import Matplotlib related modules
import matplotlib
matplotlib.use('TkAgg')
#import matplotlib.pyplot as plt
from matplotlib.backends.backend_tkagg import FigureCanvasTkAgg
from matplotlib.figure import Figure
from matplotlib.widgets import Button
#import Tkinter related modules
import Tkinter as Tk
import sys
#import Numpy
import numpy as np
#disable "Casting complex values" warning to the console. complex values/vectors
#warning is indicated on plot.
import warnings
warnings.simplefilter("ignore", np.ComplexWarning)
#define some global variables
global A,fig,ax,oldModeText,tlr,root,line1old,line2old,text1old,text2old,bSVD,svdVis
global line3old,line4old,text3old,text4old,egv1,egv2,singv1,singv2,redrawCount
global egv1txt,egv2txt,indTxt,svd1txt,svd2txt,w
#List of matrices for analysis ...one can add more...
matrixList = ['[[ 5/4, 0 ],[ 0 , 3/4]]',
'[[ 5/4, 0 ],[ 0 ,-3/4]]',
'[[ 1, 0 ],[ 0 , 1 ]] : (Identity matrix)',
'[[ 0, 1 ],[ 1 , 0 ]] : (Reflection matrix)',
'[[ 0, 1 ],[ -1 , 0 ]] : (Rotation by 90 deg)',
'[[ 1/4, 3/4],[ 4/4, 2/4]]',
'[[ 1/4, 3/4],[ 2/4, 4/4]]',
'[[ 3/4, 1/4],[ 4/4, 2/4]]',
'[[ 3/4, 1/4],[-2/4, 4/4]]',
'[[ 2/4, 4/4],[ 2/4, 4/4]]',
'[[ 2/4, 4/4],[-1/4,-2/4]]',
'[[ 6/4, 4/4],[-1/4,-2/4]]',
'[[ 0.5, 0.5],[ 0.5, 0.5]] : (Projection matrix)',
'[[ 0.8, 0.3],[ 0.2, 0.7]] : (Markov matrix)',
'randn(2,2) : (Random matrix)']
def randn(a,b):
return np.random.rand(4).reshape(a,b)
def tkQuit():
"""Stop Tk main loop and destroy figure canvas"""
global root
root.quit() # stops tk mainloop
root.destroy() # necessary to call, at least in windows
def resetAxes():
global ax, w
ax.clear() #Clear axes if already drawn
lim = np.max([1.5, np.round(np.abs(np.max(w)))])
ax.set_xlim(-lim,lim)
ax.set_ylim(-lim,lim)
ax.set_aspect('equal')
def selectMatrix(num=1):
"""To select a particular matrix"""
global A
mat = matrixList[num].partition(':')[0]
expression = 'np.matrix('+mat+')'
A = eval(expression)
reset()
def toggleSVDmode(event=None):
"""Function to determine/toggle the visibility of y and Ay"""
global bSVD, svdVis,indTxt
bSVD = not(bSVD)
svdVis = not(svdVis)
indTxt.set_visible(False)
reset()
def toggleEigenSVDvectorsVisibility(event):
"""Function to determine/toggle the visibility of the eigen and
singular vectors """
global egv1, egv2, singv1,singv2,egv1txt,egv2txt,svd1txt, svd2txt
#Toggle visibility of the eigen vectors
visible = egv1.get_visible()
egv1.set_visible(not visible and not bSVD)
egv2.set_visible(not visible and not bSVD)
egv1txt.set_visible(not visible and not bSVD)
egv2txt.set_visible(not visible and not bSVD)
#Toggle for the svd vectors
visible = singv1.get_visible()
singv1.set_visible(not visible and bSVD)
singv2.set_visible(not visible and bSVD)
svd1txt.set_visible(not visible and bSVD)
svd2txt.set_visible(not visible and bSVD)
redrawPlot(event)
drawlegend()
def reset(event=None):
"""Reset - clear the current plot, set axes, re-draw plot and legend"""
global redrawCount
redrawCount = 0
resetAxes()
drawPlot()
drawlegend()
def closeFigure(event):
"""Close the main figure"""
tkQuit()
def rotMat2D(angle,angleType='r'):
"""Return a 2D Rotation Matrix based on the input angle. The rotation is
performed in Euclidean space."""
if angleType=='d':
angle = np.radians(angle)
R = np.matrix(((np.cos(angle),-np.sin(angle)),
(np.sin(angle), np.cos(angle))))
return R
def redrawPlot(event):
"""This function is called for every mouse-click. It calculated x, Ax, y, Ay,
and re-draws it on the canvas. Since the canvas is not changed, the visibility
of older lines are set to false"""
global line1old,line2old,line3old,line4old
global text1old,text2old,text3old,text4old
global svdVis,bSVD, redrawCount
t = ax.get_window_extent().extents #returns x_0,y_0,x_1,y_1 for the axes in pixels
if ((-1 <= event.xdata <=1) and (-1 <= event.ydata <=1) and # mouse click within unit circle
(t[0]<=event.x <= t[2]) and (t[1]<= event.y<=t[3])): # mouse click within the axes (necessary)
x = np.matrix([event.xdata,event.ydata]).T
x = x/np.linalg.norm(x) #normalize the vector
Axnew = A*x
y = rotMat2D(np.pi/2)*x #perpendicular to x
y = y/np.linalg.norm(y) #normalize
Aynew = A*y
#The purpose of zordering (toggling) the scatter plot is that both red
#and blue scatter dots can be seen if they exactly overlap
zorder_b = 20
zorder_r = zorder_b + (-1)**redrawCount
redrawCount+=1
ax.scatter(x[0,0],x[1,0],c=u'r',marker='o',s=18, alpha=1.0,\
zorder=zorder_r) # x
ax.scatter(Axnew[0,0],Axnew[1,0],c=u'b',marker='o',s=18, alpha=1.0,\
zorder=zorder_b) # Ax
if bSVD:
ax.scatter(y[0,0],y[1,0],c=u'r',marker='o',s=18, alpha=1.0,\
zorder=zorder_r) # y
ax.scatter(Aynew[0,0],Aynew[1,0],c=u'b',marker='o',s=18,alpha=1.0,\
zorder=zorder_b) #Ay
#erase the old lines
line1old.set_visible(False); line2old.set_visible(False)
text1old.set_visible(False); text2old.set_visible(False)
line3old.set_visible(False); line4old.set_visible(False)
text3old.set_visible(False); text4old.set_visible(False)
#draw new lines -- x and Ax
line1new, = ax.plot([0.0,x[0,0]],[0.0,x[1,0]],c='r',aa=True)
line2new, = ax.plot([0.0,Axnew[0,0]],[0.0,Axnew[1,0]],c='b',aa=True)
text1new = ax.text( (0.0 + 0.8*tlr*x[0,0]),(0.0 + 0.8*tlr*x[1,0]),\
'$x$',fontsize=15,color='r',bbox=dict(facecolor='white',\
edgecolor='white',alpha=0.5) )
text2new = ax.text( (0.0 + tlr*Axnew[0,0]),(0.0 + tlr*Axnew[1,0]),\
'$Ax$',fontsize=15,color='b',bbox=dict(facecolor='white',\
edgecolor='white',alpha=0.5) )
#draw new lines -- y and Ay
line3new, = ax.plot([0.0,y[0,0]],[0.0,y[1,0]],c='m',aa=True)
line4new, = ax.plot([0.0,Aynew[0,0]],[0.0,Aynew[1,0]],c='g',aa=True)
text3new = ax.text( (0.0 + 0.8*tlr*y[0,0]),(0.0 + 0.8*tlr*y[1,0]),\
'$y$',fontsize=15,color='m',bbox=dict(facecolor='white',\
edgecolor='white',alpha=0.5) )
text4new = ax.text( (0.0 + tlr*Aynew[0,0]),(0.0 + tlr*Aynew[1,0]),\
'$Ay$',fontsize=15,color='g',bbox=dict(facecolor='white',\
edgecolor='white',alpha=0.5))
line3new.set_visible(svdVis);line4new.set_visible(svdVis)
text3new.set_visible(svdVis);text4new.set_visible(svdVis)
line1old,line2old = line1new,line2new
text1old,text2old = text1new,text2new
line3old,line4old = line3new,line4new
text3old,text4old = text3new,text4new
fig.canvas.draw()
def drawPlot():
"""Function to set up the lines, calculate the eigenvalue and svd """
global tlr,line1old,line2old,line3old,line4old
global text1old,text2old,text3old,text4old, oldModeText
global egv1, egv2, singv1, singv2 ,bSVD
global egv1txt, egv2txt,indTxt, svd1txt, svd2txt
global A, fig, ax, root, w #(w is make a global as it is used in resetAxis())
# update the figure text to indicate current mode (eigen/svd)
if bSVD:
currmode = 'SVD'
else:
currmode = 'Eigen'
#Calculate the eigen value and set the axis limits accordingly
w, v = np.linalg.eig(A) # w contains the eigen values, v contains the eigen vectors
resetAxes()
detA = np.linalg.det(A) #determinant
rankA = np.rank(A) #rank
traceA = np.trace(A) #trace
#complexity test
if np.sum(np.iscomplex(v)) >= 1:
complexEigenVecs = True
else:
complexEigenVecs = False
if np.sum(np.iscomplex(w)) >= 1:
complexEigenVals = True
else:
complexEigenVals = False
#fixed text to show the array
arrtext = "Matrix A = \n[[%1.3f, %1.3f],\n[%1.3f, %1.3f]]\n\ndet(A) \
= \n%1.3f\n\ntrace(A) = \n%1.3f\n\nrank(A) = \n%d" \
%(A[0,0],A[0,1],A[1,0],A[1,1],detA,traceA,rankA)
fig.text(0.013,0.20,arrtext,fontsize='medium',color='b',\
bbox=dict(facecolor='white',edgecolor='white',alpha=1.0),zorder=0)
#fixed text to indicate mode (eigen mode/svd mode)
oldModeText.set_visible(False)
modeText = fig.text(0.04,0.8,currmode,fontsize='xx-large',\
fontweight='semibold',color='#FF8000')
oldModeText = modeText
#starting lines/vectors
xstart = np.matrix([1,0]).T
ystart = np.matrix([0,1]).T #for svd mode
Axstart = np.dot(A,xstart)
Aystart = np.dot(A,ystart) #for svd mode
#Plot the columns of the matrix A
col1, = ax.plot([0.0,A[0,0]],[0.0,A[1,0]],'k--',alpha=0.6,lw='3',\
label='$col_1(A)$')
col2, = ax.plot([0.0,A[0,1]],[0.0,A[1,1]],'k--',alpha=0.4,lw='2',\
label='$col_2 (A)$')
#plot the eigen vectors (it will not be seen initially as the visibility is false)
#w, v = np.linalg.eig(A) # moved up
egv1, = ax.plot([0.0,v[0,0]],[0.0,v[1,0]],'b',lw='2',alpha=0.5,\
aa=True,label='$eigvec_1$',visible=False)
egv2, = ax.plot([0.0,v[0,1]],[0.0,v[1,1]],'r',lw='2',alpha=0.5,\
aa=True,label='$eigvec_2$',visible=False)
egv1str = "e0=%1.2f, v0=[%1.3f,%1.3f]'"%(w[0],v[0,0],v[1,0])
egv2str = "e1=%1.2f, v1=[%1.3f,%1.3f]'"%(w[1],v[0,1],v[1,1])
egv1txt = ax.text(0.01,0.06,egv1str,ha='left',color='r',\
bbox=dict(facecolor='white',edgecolor='white',alpha=1.0),\
visible=False,transform = ax.transAxes)
egv2txt = ax.text(0.01,0.02,egv2str,ha='left',color='r',\
bbox=dict(facecolor='white',edgecolor='white',alpha=1.0),\
visible=False,transform = ax.transAxes)
#calculate the svd
U,S,V = np.linalg.svd(A)
#complexity test
if np.sum(np.iscomplex(U)) >= 1:
complexSingVecs = True
else:
complexSingVecs = False
if np.sum(np.iscomplex(S)) >= 1:
complexSingVals = True
else:
complexSingVals = False
#plot the svd (it will not be seen initially as the visibility is false)
singv1, = ax.plot([0.0,U[0,0]],[0.0,U[1,0]],'g--',lw='2',alpha=0.5,\
aa=True,label='$singvec_1$',visible=False)
singv2, = ax.plot([0.0,U[0,1]],[0.0,U[1,1]],'m--',lw='2',alpha=0.5,\
aa=True,label='$singvec_2$',visible=False)
svd1str = "s0=%1.2f, u0=[%1.3f,%1.3f]'"%(S[0],U[0,0],U[1,0])
svd2str = "s1=%1.2f, u1=[%1.3f,%1.3f]'"%(S[1],U[0,1],U[1,1])
svd1txt = ax.text(0.01,0.06,svd1str,ha='left',color='r',\
bbox=dict(facecolor='white',edgecolor='white',alpha=1.0),visible=False,\
transform = ax.transAxes)
svd2txt = ax.text(0.01,0.02,svd2str,ha='left',color='r',\
bbox=dict(facecolor='white',edgecolor='white',alpha=1.0),visible=False,\
transform = ax.transAxes)
#lines related to just the eigen vectors
line1, = ax.plot([0.0,xstart[0,0]],[0.0,xstart[1,0]],aa=True,c='r')
line2, = ax.plot([0.0,Axstart[0,0]],[0.0,Axstart[1,0]],aa=True,c='b')
text1 = ax.text( (0.0 + 0.8*tlr*xstart[0,0]),(0.0 + 0.8*tlr*xstart[1,0]),\
'$x$',fontsize=15,color='r',bbox=dict(facecolor='white',edgecolor='white',\
alpha=0.5) )
text2 = ax.text( (0.0 + tlr*Axstart[0,0]),(0.0 + tlr*Axstart[1,0]),'$Ax$',\
fontsize=15,color='b',bbox=dict(facecolor='white',edgecolor='white',alpha=0.5))
line1old,line2old = line1,line2
text1old,text2old = text1,text2
#lines related to just the svd vectors (depending on the svdVis)
line3, = ax.plot([0.0,ystart[0,0]],[0.0,ystart[1,0]],aa=True,c='m',\
visible=svdVis)
line4, = ax.plot([0.0,Aystart[0,0]],[0.0,Aystart[1,0]],aa=True,c='g',\
visible=svdVis)
text3 = ax.text( (0.0 + 0.8*tlr*ystart[0,0]),(0.0 + 0.8*tlr*ystart[1,0]),\
'$y$',fontsize=15,color='m',bbox=dict(facecolor='white',edgecolor='white',\
alpha=0.5),visible=svdVis)
text4 = ax.text( (0.0 + tlr*Aystart[0,0]),(0.0 + tlr*Aystart[1,0]),'$Ay$',\
fontsize=15,color='g',bbox=dict(facecolor='white',edgecolor='white',\
alpha=0.5),visible=svdVis)
line3old,line4old = line3,line4
text3old,text4old = text3,text4
#Text to indicate complex/real nature of vectors and values
if complexEigenVals and not bSVD:
comValTxt = ax.text(0- 0.5*ax.get_xlim()[1],0.5*ax.get_ylim()[1],\
"Complex eigen values",ha='left',\
color='y',fontsize='large',fontweight='bold',alpha=0.4)
if complexEigenVecs and not bSVD:
comVecTxt = ax.text(0- 0.5*ax.get_xlim()[1],0.4*ax.get_ylim()[1],\
"Complex eigen vectors",ha='left',\
color='y',fontsize='large',fontweight='bold',alpha=0.4)
if complexSingVals and bSVD:
comValTxt = ax.text(0- 0.5*ax.get_xlim()[1],0.5*ax.get_ylim()[1],\
"Complex singular values",ha='left',\
color='y',fontsize='large',fontweight='bold',alpha=0.4)
if complexSingVecs and bSVD:
comVecTxt = ax.text(0- 0.5*ax.get_xlim()[1],0.4*ax.get_ylim()[1],\
"Complex singular vectors",ha='left',\
color='y',fontsize='large',fontweight='bold',alpha=0.4)
# Text to indicate goal
if not bSVD:
indStr = "Make A*x parallel to x ." #don't change space
else:
indStr = "Make A*x perpendicular to A*y ." #don't change space
indTxt = fig.text(0.3,0.03,indStr,ha='left',color='g',fontsize='large',\
fontweight='bold',bbox=dict(facecolor='white',edgecolor='white',alpha=1.0),\
visible=True)
def drawlegend():
ax.legend(loc='upper center', bbox_to_anchor=(0.5, 1.1),ncol=3, \
fancybox=True, shadow=True)
def eigshow(matrix = None):
"""Main function to plot eigshow.
Usage: eigshow()
or
eigshow(A)
where, A is a Numpy 2x2 matrix or array.
When no aguments are passed to eigshow(), it starts with a default matrix
A = np.matrix([[1/4,3/4],[1,2/4]]) and one can choose several other matrices
using the "Select Matrix" menu on the top menu panel.
When A is given, eigshow starts with the given matrix."""
global line1old,line2old,text1old,text2old,line3old,line4old,text3old
global text4old,egv1,egv2,singv1,singv2,egv1txt,egv2txt,indTxt
global svd1txt, svd2txt, bSVD, svdVis, oldModeText, tlr, redrawCount
global root, A, fig, ax
bSVD = False #svd mode or not (initially set to eigen mode)
svdVis = False #Visibility of lines for svd mode (not visible in eigen mode)
tlr = 0.75 #text on line position ratio
redrawCount = 0
if matrix == None:
#The matrix (initial matrix)
A = np.matrix([[1/4,3/4],[1,2/4]]) #matrixList[4]
else:
A = np.matrix(matrix)
root = Tk.Tk()
root.wm_title('eigshow')
#Create a toplevel menu (for selecting matrices)
menubar = Tk.Menu(root)
#Create the Matrix pulldown menu, and add it to the menubar
filemenu = Tk.Menu(menubar,tearoff=0)
for i, mat in enumerate(matrixList):
expression = \
'filemenu.add_command(label=mat,command=lambda: selectMatrix({}))'.format(i)
eval(expression)
menubar.add_cascade(label="Select Matrix",menu=filemenu)
#Display the menu
root.config(menu=menubar)
#Create a figure and an axes within it
fig = Figure(facecolor='w')
ax = fig.add_subplot(111)
# a tk drawing area
canvas = FigureCanvasTkAgg(fig,master=root)#no resize callback as of now
canvas.get_tk_widget().pack(side=Tk.BOTTOM)
#Connect redrawPlot callback function to mouse-click event
fig.canvas.mpl_connect('button_press_event',redrawPlot)
#buttons on the right on main figure
ax_reset = fig.add_axes([0.83, 0.70, 0.16, 0.12])
b_reset = Button(ax_reset, 'Reset',color='0.95',hovercolor='0.85')
b_reset.on_clicked(reset)
ax_eigen_svd = fig.add_axes([0.83, 0.55, 0.16, 0.12])
b_eigen_svd =Button(ax_eigen_svd,'Eigen/SVD',color='0.95',hovercolor='0.85')
b_eigen_svd.on_clicked(toggleSVDmode)
ax_showvecs = fig.add_axes([0.83, 0.40, 0.16, 0.12])
b_showvecs = Button(ax_showvecs, 'Show\nEigen/Singular\nvectors',\
color='0.95', hovercolor='0.85')
b_showvecs.on_clicked(toggleEigenSVDvectorsVisibility)
ax_closeFig = fig.add_axes([0.83, 0.25, 0.16, 0.12])
b_closeFig = Button(ax_closeFig,'Close',color='0.95',hovercolor='0.85')
b_closeFig.on_clicked(closeFigure)
#Initialize some of the objects (lines, texts, etc)
oldModeText = fig.text(0.01,0.8,'dummytext',fontsize='large') #dummy text
#Start rendering the plot
drawPlot()
drawlegend()
#Draw the plot on the canvas
canvas.show()
Tk.mainloop()
if __name__ == '__main__':
eigshow()
Lastly, here is another picture of the eigshow program in the SVD mode.
Indranil's world 