将对称矩阵(2D阵列)的上/下三角形部分转换为1D阵列并将其返回到2D格式

Question

在这个问题阐述了如何访问lower和upper给定矩阵的triagular部分,说:

m = np.matrix([[11, 12, 13],
               [21, 22, 23],
               [31, 32, 33]])

在这里,我需要在一维数组中转换矩阵,可以这样做:

indices = np.triu_indices_from(m)
a = np.asarray( m[indices] )[-1]
#array([11, 12, 13, 22, 23, 33])

在进行大量计算后a,更改其值,它将用于填充对称的2D数组:

new = np.zeros(m.shape)
for i,j in enumerate(zip(*indices)):
    new[j]=a[i]
    new[j[1],j[0]]=a[i]

返回:

array([[ 11.,  12.,  13.],
       [ 12.,  22.,  23.],
       [ 13.,  23.,  33.]])

有没有更好的方法来实现这一目标？更具体地说,避免Python循环重建2D数组？

Answer 1

将向量放回二维对称数组的最快和最聪明的方法是这样做：

情况 1：无偏移 (k=0) 即上三角形部分包括对角线

import numpy as np

X = np.array([[1,2,3],[4,5,6],[7,8,9]])
#array([[1, 2, 3],
#       [4, 5, 6],
#       [7, 8, 9]])

#get the upper triangular part of this matrix
v = X[np.triu_indices(X.shape[0], k = 0)]
print(v)
# [1 2 3 5 6 9]

# put it back into a 2D symmetric array
size_X = 3
X = np.zeros((size_X,size_X))
X[np.triu_indices(X.shape[0], k = 0)] = v
X = X + X.T - np.diag(np.diag(X))
#array([[1., 2., 3.],
#       [2., 5., 6.],
#       [3., 6., 9.]])

即使numpy.array您使用numpy.matrix.

情况 2：偏移量 (k=1) 即上三角形部分不包括对角线

import numpy as np

X = np.array([[1,2,3],[4,5,6],[7,8,9]])
#array([[1, 2, 3],
#       [4, 5, 6],
#       [7, 8, 9]])

#get the upper triangular part of this matrix
v = X[np.triu_indices(X.shape[0], k = 1)] # offset
print(v)
# [2 3 6]

# put it back into a 2D symmetric array
size_X = 3
X = np.zeros((size_X,size_X))
X[np.triu_indices(X.shape[0], k = 1)] = v
X = X + X.T
#array([[0., 2., 3.],
#       [2., 0., 6.],
#       [3., 6., 0.]])

Answer 2

你只想形成一个对称阵列吗？您可以完全跳过对角线索引.

m=np.array(m)
inds = np.triu_indices_from(m,k=1)
m[(inds[1], inds[0])] = m[inds]

m

array([[11, 12, 13],
       [12, 22, 23],
       [13, 23, 33]])

从以下位置创建对称数组:

new = np.zeros((3,3))
vals = np.array([11, 12, 13, 22, 23, 33])
inds = np.triu_indices_from(new)
new[inds] = vals
new[(inds[1], inds[0])] = vals
new
array([[ 11.,  12.,  13.],
       [ 12.,  22.,  23.],
       [ 13.,  23.,  33.]])

Answer 3

您可以使用数组创建例程（例如numpy.triu、numpy.tril和numpy.diag ）从三角形创建对称矩阵。这是一个简单的 3x3 示例。

a = np.array([[1,2,3],[4,5,6],[7,8,9]])
array([[1, 2, 3],
       [4, 5, 6],
       [7, 8, 9]])

a_triu = np.triu(a, k=0)
array([[1, 2, 3],
       [0, 5, 6],
       [0, 0, 9]])

a_tril = np.tril(a, k=0)
array([[1, 0, 0],
       [4, 5, 0],
       [7, 8, 9]])

a_diag = np.diag(np.diag(a))
array([[1, 0, 0],
       [0, 5, 0],
       [0, 0, 9]])

添加转置并减去对角线：

a_sym_triu = a_triu + a_triu.T - a_diag
array([[1, 2, 3],
       [2, 5, 6],
       [3, 6, 9]])

a_sym_tril = a_tril + a_tril.T - a_diag
array([[1, 4, 7],
       [4, 5, 8],
       [7, 8, 9]])