import numpy as np
import numpy.linalg as alg

P = np.array([[1,2,1],[1,3,2],[1,3,3]])
print("det(P)=",alg.det(P))
P1 = alg.inv(P)

U = np.array([[-2,4,-2],[-3,6,-3],[-3,6,-3]])
V = np.array([[0,-2,2],[0,-5,5],[0,-5,5]])

U1 = np.dot(P1,np.dot(U,P))
V1 = np.dot(P1,np.dot(V,P))
print("P^{-1}UPp=",U1)
print("P^{-1}VP=",V1)

for a in [0,0.5,1] :
    W = U+a*V
    print("W(",a,")tW(",a,")=",np.dot(W,W.T))
    print("tW(",a,")W(",a,")=",np.dot(W.T,W))