What is the best way to compute the trace of a matrix product in numpy?

Question

If I have numpy arrays A and B, then I can compute the trace of their matrix product with:

tr = numpy.linalg.trace(A.dot(B))
         


        

Answer 1

Note that one slight variant is to take the dot product of the vectorized matrices. In python, vectorization is done using .flatten('F'). It's slightly slower than taking the sum of the Hadamard product, on my computer, so it's a worse solution than wflynny's , but I think it's kind of interesting, since it can be more intuitive, in some situations, in my opinion. For example, personally I find that for the matrix normal distribution, the vectorized solution is easier for me to understand.

Speed comparison, on my system:

import numpy as np
import time

N = 1000

np.random.seed(123)
A = np.random.randn(N, N)
B = np.random.randn(N, N)

tart = time.time()
for i in range(10):
    C = np.trace(A.dot(B))
print(time.time() - start, C)

start = time.time()
for i in range(10):
    C = A.flatten('F').dot(B.T.flatten('F'))
print(time.time() - start, C)

start = time.time()
for i in range(10):
    C = (A.T * B).sum()
print(time.time() - start, C)

start = time.time()
for i in range(10):
    C = (A * B.T).sum()
print(time.time() - start, C)

Result:

6.246593236923218 -629.370798672
0.06539678573608398 -629.370798672
0.057890892028808594 -629.370798672
0.05709719657897949 -629.370798672