Bruteforce matrix function relationships

properties.py

from sympy import *
import itertools

def M(a, f1, f2, f3, f4):
    a1, a2, a3, a4 = a.values()

    M1 = f1 * Matrix([[0,-a3, a2,0],[a3, 0, -a1,0],[-a2, a1, 0,0],[0,0,0,0]])
    M2 = f2 * Matrix([[0,0, 0,0],[0, 0, 0,0],[0, 0, 0,0],[a1,a2,a3,0]])
    M3 = f3 * Matrix([[0,0, 0,a1],[0, 0, 0,a2],[0, 0, 0,a3],[0,0,0,0]])
    M4 = f4 * a4 * eye(4)

    return f1*M1 + f2*M2 + f3*M3 + f4*M4

def W(a):
    return M(a, -1,-1,1,1)

def Q(a):
    return M(a, 1,-1,1,1)

def U(a):
    return M(a, -1,1,1,-1)

def get_properties(expr,a,b):
    Z = zeros(4)
    it = list(itertools.product([-1,1],repeat=4))
    Ma = [M(a,i,j,k,l) for i,j,k,l in it]
    Mb = [M(b,i,j,k,l) for i,j,k,l in it]
    for f, F in enumerate(Ma):
        for g, G in enumerate(Mb):
            diff1 = expr - F*G
            diff2 = expr - F*G.T
            diff3 = expr - F.T*G
            diff4 = expr - F.T*G.T

            if diff1.equals(Z) or diff2.equals(Z) or diff3.equals(Z) or diff4.equals(Z):
                i,j,k,l = it[f]
                I,J,K,L = it[g]
                if (i,j,k,l) == (-1,-1,1,1):
                    print "W",
                elif (i,j,k,l) == (1,-1,1,1):
                    print "Q",
                elif (i,j,k,l) == (-1,1,1,-1):
                    print "U",
                else:
                    print (i,j,k,l),

                if (I,J,K,L) == (-1,-1,1,1):
                    print "W",
                elif (I,J,K,L) == (1,-1,1,1):
                    print "Q",
                elif (I,J,K,L) == (-1,1,1,-1):
                    print "U",
                else:
                    print (I,J,K,L)

                if diff1.equals(Z):
                    print 1
                if diff2.equals(Z):
                    print 2
                if diff3.equals(Z):
                    print 3
                if diff4.equals(Z):
                    print 4



a1, a2, a3, a4, b1, b2, b3, b4 = symbols('a1 a2 a3 a4 b1 b2 b3 b4')
a = Matrix([[a1,a2,a3,a4]]).T
b = Matrix([[b1,b2,b3,b4]]).T

for f1 in "Q","W","U":
    if f1 == "Q":
        A1 = Q
    elif f1 == "W":
        A1 = W
    else:
        A1 = U
    for f2 in "Q","W","U":
        if f2 == "Q":
            B1 = Q
        elif f2 == "W":
            B1 = W
        else:
            B1 = U
        for t1 in True,False:
            for t2 in True,False:
                for e1, e2 in ["aa","ab","ba"]:
                    if e1 == "a":
                        A2 = A1(a)
                    else:
                        A2 = A1(b)
                    if e2 == "a":
                        B2 = B1(a)
                    else:
                        B2 = B1(b)
                    if t1:
                        A3 = A2.T
                    else:
                        A3 = A2
                    if t2:
                        B3 = B2.T
                    else:
                        B3 = B2
                    print f1, f2, t1, t2, e1, e2
                    get_properties(A3*B3, a, b)