import sympy as sp
import numpy as np
import matplotlib.pyplot as plt

#Defining the matrix

A3 = sp.Matrix([[(1/11),(13/28),(0),(0),(14/22)],[(15/22),(1/4),(7/30),(0),(0)],
                [(0),(8/28),(1/6),(10/34),(0)],[(0),(0),(18/30),(1/34),(5/22)],[(5/22),(0),(0),(23/34),(3/22)]])
#Finding the eigenvalues and eigenvectors
#note that Python crashes due to the imaginary eigenvectors.
A3.eigenvals()
A3.eigenvects()

#Define the population distribution vectors and multiply by a large power of the matrix.
v = (sp.Matrix([[0.25,0.15,0.2,0.2,0.2]])).transpose()
u = (sp.Matrix([[0.1,0.2,0.1,0.5,0.1]])).transpose()
k = (sp.Matrix([[0.2,0.1,0.3,0.4,0]])).transpose()

print((A3**10000)*v)
print((A3**10000)*u)
print((A3**10000)*k)

"""
Question 2
"""
#Define the matrix, apply the Gram Schmidt algorithm, and then define Q according to the values output by the Gram Schmidt algorithm.
#Then, find R.
A1 = sp.Matrix([[2,3],[0,-2]])

gsA1=[sp.Matrix([[2,0]]),sp.Matrix([[3,-2]])]
print(sp.GramSchmidt(gsA1))

Q1 = sp.Matrix([[2,0],[0,-2]])
R1=(Q1.transpose())*A1
(R1)

A2 = sp.Matrix([[1,1,0],[1,0,1],[0,1,1]])

gsA2=[sp.Matrix([[1,1,0]]),sp.Matrix([[1,0,1]]),sp.Matrix([[0,1,1]])]
print(sp.GramSchmidt(gsA2))

Q2 = sp.Matrix([[1,(1/2),(-2/3)],[1,(-1/2),(2/3)],[0,1,(2/3)]])
R2=(Q2.transpose())*A2
(R2)

#Code for creating the arrow diagram.

V = np.array([[1,0],[0,1],[4,3]])
origin = [0], [0] # origin point

plt.quiver(*origin, V[:,0], V[:,1], color=['r','b','g'], scale=10)
plt.show()

#Using python to generate the column spaces for the last part.

print("A1:",A1.columnspace())
print("Q1:",Q1.columnspace())
print("Q1 Inverse:", (sp.Matrix([[2,0],[0,-2]])).transpose())
print("A2:",A2.columnspace())
print("Q2:",Q2.transpose())