python - Numpy, dot products on multidimensional arrays -

i have doubts on numpy.dot product.

i define matrix 6x6 like:

c=np.zeros((6,6)) c[0,0], c[1,1], c[2,2] = 129.5, 129.5, 129.5 c[3,3], c[4,4], c[5,5] = 25, 25, 25 c[0,1], c[0,2] = 82, 82 c[1,0], c[1,2] = 82, 82 c[2,0], c[2,1] = 82, 82 

then recast in 4-rank tensor using multidimensional array

def long2short(m, n):     """     given 2 indices m , n of stiffness tensor function     return index of voigt matrix     = long2short(m,n)     """     if m == n:         = m     elif (m == 1 , n == 2) or (m == 2 , n == 1):         = 3     elif (m == 0 , n == 2) or (m == 2 , n == 0):         = 4     elif (m == 0 , n == 1) or (m == 1 , n == 0):         = 5           return  c=np.zeros((3,3,3,3)) m in range(3):     n in range(3):         o in range(3):             p in range(3):                 = long2short(m, n)                 j = long2short(o, p)                 c[m, n, o, p] = c[i, j] 

and change coordinate reference system of tensor using rotation matrix define like:

q=np.array([[sqrt(2.0/3), 0, 1.0/sqrt(3)], [-1.0/sqrt(6), 1.0/sqrt(2), 1.0/sqrt(3)], [-1.0/sqrt(6), -1.0/sqrt(2), 1.0/sqrt(3)]])         qt = q.transpose() 

the matrix orthogonal (althought numerical precision not perfect):

in [157]: np.dot(q, qt) out[157]:  array([[  1.00000000e+00,   4.28259858e-17,   4.28259858e-17],        [  4.28259858e-17,   1.00000000e+00,   2.24240114e-16],        [  4.28259858e-17,   2.24240114e-16,   1.00000000e+00]]) 

but why if perform:

in [158]: a=np.dot(q,qt) in [159]: c_mat=np.dot(a, c) in [160]: a1 = np.dot(qt, c) in [161]: c_mat1=np.dot(q, a1) 

i expected value c_mat (=c) not c_mat1? there subtility use dot on multidimensional arrays?

the issue np.dot(a,b) multidimensional arrays makes dot product of last dimension of a second-to-last dimension of b:

np.dot(a,b) == np.tensordot(a, b, axes=([-1],[2])) 

as see, not work matrix multiplication multidimensional arrays. using np.tensordot() allows control in axes each input want perform dot product. example, same result in c_mat1 can do:

c_mat1 = np.tensordot(q, a1, axes=([-1],[0])) 

which forcing matrix multiplication-like behavior.