numpy.linal . linalgerror:在np.dot中，数组的最后两个维度必须是正方形的

Question

A = np.array([[2,-1],[-1,2],[0,-1]])
q, r = np.linalg.qr(A, 'complete')
print(q)
print(r)
# The line below will raise error: numpy.linalg.LinAlgError: Last 2 dimensions of the array must be square
print(np.dot(np.linalg.inv(r), q.T))
# Mathmatically, I think it should output something similar to the following code, if im not wrong. The following code should be able to run
print(np.dot(np.linalg.inv(np.dot(A.T, A)), A.T)) 

我想让这段代码工作,但得到这个错误:

numpy.linalg。LinAlgError:数组的最后两个维度必须是广场

我知道第二个矩阵不是平方的，但是无论如何我可以使这个代码工作吗?例如，有没有一种方法可以绕过这个或者把矩阵变成平方?

Answer 1

In [117]: q
Out[117]: 
array([[-0.89442719, -0.35856858,  0.26726124],
[ 0.4472136 , -0.71713717,  0.53452248],
[-0.        ,  0.5976143 ,  0.80178373]])
In [118]: r        # same shape as A, (3,2) as per docs
Out[118]: 
array([[-2.23606798,  1.78885438],
[ 0.        , -1.67332005],
[ 0.        ,  0.        ]])

r没有0的行是可逆的:

In [120]: np.linalg.inv(r[:2])
Out[120]: 
array([[-0.4472136 , -0.47809144],
[-0.        , -0.5976143 ]])

我们可以用q的一部分来dot它:

In [130]: np.dot(np.linalg.inv(r[:2]),q.T[:2])
Out[130]: 
array([[ 0.57142857,  0.14285714, -0.28571429],
[ 0.21428571,  0.42857143, -0.35714286]])

匹配:

In [131]: np.dot(np.linalg.inv(np.dot(A.T, A)), A.T)
Out[131]: 
array([[ 0.57142857,  0.14285714, -0.28571429],
[ 0.21428571,  0.42857143, -0.35714286]])

pinv

In [5]: np.linalg.pinv(r)
Out[5]: 
array([[-0.4472136 , -0.47809144,  0.        ],
[ 0.        , -0.5976143 ,  0.        ]])
In [6]: np.dot(np.linalg.pinv(r),q.T)
Out[6]: 
array([[ 0.57142857,  0.14285714, -0.28571429],
[ 0.21428571,  0.42857143, -0.35714286]])