多维数组上的Numpy,dot产品

Question

我对numpy.dot产品有些怀疑.

我定义了一个矩阵6x6,如:

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

然后我通过使用多维数组以4级张量重新制作它

def long2short(m, n):
    """
    Given two indices m and n of the stiffness tensor the function
    return i the index of the Voigt matrix
    i = long2short(m,n)
    """
    if m == n:
        i = m
    elif (m == 1 and n == 2) or (m == 2 and n == 1):
        i = 3
    elif (m == 0 and n == 2) or (m == 2 and n == 0):
        i = 4
    elif (m == 0 and n == 1) or (m == 1 and n == 0):
        i = 5      
    return i

c=np.zeros((3,3,3,3))
for m in range(3):
    for n in range(3):
        for o in range(3):
            for p in range(3):
                i = long2short(m, n)
                j = long2short(o, p)
                c[m, n, o, p] = C[i, j]

然后我想通过使用我定义的旋转矩阵来改变张量的坐标参考系统:

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()

矩阵是正交的(虽然数值精度不完美):

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]])

但是为什么我这样做呢:

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)

我得到c_mat(= c)的期望值,但c_mat1没有？在多维数组上使用点有一些不足之处吗？

Answer 1

问题是,np.dot(a,b)对于多维数组,最后一个维度的点积与倒数a第二个维度b:

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

如您所见,它不能用作多维数组的矩阵乘法.使用np.tensordot()允许您控制axes要从中执行点积的每个输入.例如,要获得相同的结果,c_mat1您可以执行以下操作:

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

这就是强制类似矩阵乘法的行为.