M_i . M = M_o

M_i\cdot M = M_o.py

import numpy as np

np.random.seed(64)

M_i = np.random.random((3, 3))

M_o = np.random.random((3, 3))

# M . M_i = M_o
# M . M_i . M_i^{-1} = M_o . M_i^{-1}
# M = M_o . M_i^{-1}

M = np.dot(M_o, np.linalg.inv(M_i))

XYZ = np.random.random((3, 1))

print(XYZ)

print(np.dot(M, np.dot(M_i, XYZ)))

print(np.dot(M_o, XYZ))

# M_i . M = M_o
# M_i^{-1} . M_i . M = M_i^{-1} . M_o
# M = M_i^{-1} . M_o

M = np.dot(np.linalg.inv(M_i), M_o)

print(np.dot(M_i, np.dot(M, XYZ)))

print(np.dot(M_o, XYZ))

# [[ 0.22444174]
#  [ 0.39830987]
#  [ 0.6195008 ]]
# [[ 0.71475756]
#  [ 0.21555621]
#  [ 0.44902374]]
# [[ 0.71475756]
#  [ 0.21555621]
#  [ 0.44902374]]
# [[ 0.71475756]
#  [ 0.21555621]
#  [ 0.44902374]]
# [[ 0.71475756]
#  [ 0.21555621]
#  [ 0.44902374]]