在Python中获取矩阵/列表列表中的所有对角线

Question

我正在寻找一种 Pythonic 方法来获取（方形）矩阵的所有对角线，表示为列表的列表。

假设我有以下矩阵：

matrix = [[-2,  5,  3,  2],
          [ 9, -6,  5,  1],
          [ 3,  2,  7,  3],
          [-1,  8, -4,  8]]

那么大对角线就简单了：

l = len(matrix[0])
print([matrix[i][i] for i in range(l)])              # [-2, -6, 7,  8]
print([matrix[l-1-i][i] for i in range(l-1,-1,-1)])  # [ 2,  5, 2, -1]

但是我很难想出一种方法来生成所有对角线。我正在寻找的输出是：

[[-2], [9, 5], [3,-6, 3], [-1, 2, 5, 2], [8, 7, 1], [-4, 3], [8],
 [2], [3,1], [5, 5, 3], [-2, -6, 7, 8], [9, 2, -4], [3, 8], [-1]]

Answer 1

numpy 中可能有比下面更好的方法，但我还不太熟悉：

import numpy as np

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

diags = [matrix[::-1,:].diagonal(i) for i in range(-3,4)]
diags.extend(matrix.diagonal(i) for i in range(3,-4,-1))
print [n.tolist() for n in diags]

输出

[[-2], [9, 5], [3, -6, 3], [-1, 2, 5, 2], [8, 7, 1], [-4, 3], [8], [2], [3, 1], [5, 5, 3], [-2, -6, 7, 8], [9, 2, -4], [3, 8], [-1]]

---

编辑：更新为适用于任何矩阵大小。

import numpy as np

# Alter dimensions as needed
x,y = 3,4

# create a default array of specified dimensions
a = np.arange(x*y).reshape(x,y)
print a
print

# a.diagonal returns the top-left-to-lower-right diagonal "i"
# according to this diagram:
#
#  0  1  2  3  4 ...
# -1  0  1  2  3
# -2 -1  0  1  2
# -3 -2 -1  0  1
#  :
#
# You wanted lower-left-to-upper-right and upper-left-to-lower-right diagonals.
#
# The syntax a[slice,slice] returns a new array with elements from the sliced ranges,
# where "slice" is Python's [start[:stop[:step]] format.

# "::-1" returns the rows in reverse. ":" returns the columns as is,
# effectively vertically mirroring the original array so the wanted diagonals are
# lower-right-to-uppper-left.
#
# Then a list comprehension is used to collect all the diagonals.  The range
# is -x+1 to y (exclusive of y), so for a matrix like the example above
# (x,y) = (4,5) = -3 to 4.
diags = [a[::-1,:].diagonal(i) for i in range(-a.shape[0]+1,a.shape[1])]

# Now back to the original array to get the upper-left-to-lower-right diagonals,
# starting from the right, so the range needed for shape (x,y) was y-1 to -x+1 descending.
diags.extend(a.diagonal(i) for i in range(a.shape[1]-1,-a.shape[0],-1))

# Another list comp to convert back to Python lists from numpy arrays,
# so it prints what you requested.
print [n.tolist() for n in diags]

输出

[[ 0  1  2  3]
 [ 4  5  6  7]
 [ 8  9 10 11]]

[[0], [4, 1], [8, 5, 2], [9, 6, 3], [10, 7], [11], [3], [2, 7], [1, 6, 11], [0, 5, 10], [4, 9], [8]]

Answer 2

我遇到了这个问题的另一个有趣的解决方案。 通过查看 x 和 y 的组合，可以立即发现行、列、前向和后向对角线。

Column = x     Row = y        F-Diag = x+y   B-Diag = x-y     B-Diag` = x-y-MIN 
  | 0  1  2      | 0  1  2      | 0  1  2      | 0  1  2        | 0  1  2     
--|---------   --|---------   --|---------   --|---------     --|---------    
0 | 0  1  2    0 | 0  0  0    0 | 0  1  2    0 | 0  1  2      0 | 2  3  4     
1 | 0  1  2    1 | 1  1  1    1 | 1  2  3    1 |-1  0  1      1 | 1  2  3     
2 | 0  1  2    2 | 2  2  2    2 | 2  3  4    2 |-2 -1  0      2 | 0  1  2     

从图中您可以看到，使用这些方程式可以唯一地识别每个对角线和轴。从每个表中获取每个唯一编号并为该标识符创建一个容器。

请注意，后向对角线已偏移以从零索引开始，并且前向对角线的长度始终等于后向对角线的长度。

test = [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]

max_col = len(test[0])
max_row = len(test)
cols = [[] for _ in range(max_col)]
rows = [[] for _ in range(max_row)]
fdiag = [[] for _ in range(max_row + max_col - 1)]
bdiag = [[] for _ in range(len(fdiag))]
min_bdiag = -max_row + 1

for x in range(max_col):
    for y in range(max_row):
        cols[x].append(test[y][x])
        rows[y].append(test[y][x])
        fdiag[x+y].append(test[y][x])
        bdiag[x-y-min_bdiag].append(test[y][x])

print(cols)
print(rows)
print(fdiag)
print(bdiag)

哪个会打印

[[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 11, 12]]
[[1, 4, 7, 10], [2, 5, 8, 11], [3, 6, 9, 12]]
[[1], [2, 4], [3, 5, 7], [6, 8, 10], [9, 11], [12]]
[[10], [7, 11], [4, 8, 12], [1, 5, 9], [2, 6], [3]]

Answer 3

从向上和向右倾斜的对角线开始。

如果 (x,y) 是矩阵内的直角坐标，您想要转换到/从坐标方案 (p,q) 转换，其中 p 是对角线的编号，q 是沿对角线的索引。 （所以 p=0 是 [-2] 对角线，p=1 是 [9,5] 对角线，p=2 是 [3,-6,3] 对角线，依此类推。）

要将 (p,q) 转换为 (x,y)，您可以使用：

x = q
y = p - q

尝试插入 p 和 q 的值，看看它是如何工作的。

现在您只需循环...对于从 0 到 2N-1 的 p，以及从 max(0, p-N+1) 到 min(p, N-1) 的 q。将 p,q 转换为 x,y 并打印。

然后对于其他对角线，重复循环但使用不同的变换：

x = N - 1 - q
y = p - q

（这实际上只是左右翻转矩阵。）

对不起，我实际上并没有用 Python 编写代码。 :-)

Answer 4

这是给Moe的，谁问过a similar question。

我首先制作简单的函数来复制任何矩形矩阵的行或列。

def get_rows(grid):
    return [[c for c in r] for r in grid]

def get_cols(grid):
    return zip(*grid)

使用这两个函数，然后我通过在每行的开始/结束处添加一个递增/递减缓冲区来获得对角线。然后我得到这个缓冲网格的列，然后删除每列上的缓冲区。即）

1 2 3    |X|X|1|2|3|    | | |1|2|3|
4 5 6 => |X|4|5|6|X| => | |4|5|6| | => [[7],[4,8],[1,5,9],[2,6],[3]]
7 8 9    |7|8|9|X|X|    |7|8|9| | |

.

def get_backward_diagonals(grid):
    b = [None] * (len(grid) - 1)
    grid = [b[i:] + r + b[:i] for i, r in enumerate(get_rows(grid))]
    return [[c for c in r if c is not None] for r in get_cols(grid)]

def get_forward_diagonals(grid):
    b = [None] * (len(grid) - 1)
    grid = [b[:i] + r + b[i:] for i, r in enumerate(get_rows(grid))]
    return [[c for c in r if c is not None] for r in get_cols(grid)]

Answer 5

我最近重新发明了这个轮子。这是一种易于重用/扩展的方法，可以在方形列表列表中找到对角线：

def get_diagonals(grid, bltr = True):
  dim = len(grid)
  assert dim == len(grid[0])
  return_grid = [[] for total in xrange(2 * len(grid) - 1)]
  for row in xrange(len(grid)):
    for col in xrange(len(grid[row])):
      if bltr: return_grid[row + col].append(grid[col][row])
      else:    return_grid[col - row + (dim - 1)].append(grid[row][col])
  return return_grid

假设列表索引：

00 01 02 03

10 11 12 13

20 21 22 23

30 31 32 33

然后设置bltr = True（默认），返回从左下角到右上角的对角线，即

00           # row + col == 0
10 01        # row + col == 1
20 11 02     # row + col == 2
30 21 12 03  # row + col == 3
31 22 13     # row + col == 4
32 23        # row + col == 5
33           # row + col == 6

设置bltr = False，返回从左下角到右上角的对角线，即

30            # (col - row) == -3
20 31         # (col - row) == -2
10 21 32      # (col - row) == -1
00 11 22 33   # (col - row) == 0
01 12 23      # (col - row) == +1
02 13         # (col - row) == +2
03            # (col - row) == +3

Here's a runnable version 使用 OP 的输入矩阵。

Answer 6

我想现在有一种更简单的方法可以做到这一点。 （但只有在您已经熟悉上述答案时才使用它）。

from collections import defaultdict

有一个名为 defaultdict 的方法，它是从 collections 模块中导入的，如果您不知道要拥有的密钥，可以使用它来创建字典。

我们在这些情况下使用它：

• 如果您不知道键但想为特定键分配一些值。
• 如果字典中不存在键，普通字典会引发 keyerror。但这不会（如果需要，您可以为其分配一些功能）

导入后，可以运行以下代码进行检查。

rows,cols = 3,3
matrix = [[1, 2, 3],
          [4, 5, 6],
          [7, 8, 9]]

diagonal1 = defaultdict(list) # For the top right to bottom left
diagonal2 = defaultdict(list) # For the top left to bottom right
for i in range(rows):
    for j in range(cols):
        diagonal1[i-j].append(matrix[i][j])
        diagonal2[i+j].append(matrix[i][j])
print(diagonal1,'\n',diagonal2)

list 参数将为该特定键创建一个值列表。

输出如下：

defaultdict(<class 'list'>, {0: [1, 5, 9], -1: [2, 6], -2: [3], 1: [4, 8], 2: [7]}) 
defaultdict(<class 'list'>, {0: [1], 1: [2, 4], 2: [3, 5, 7], 3: [6, 8], 4: [9]})

现在你可以随意使用两条对角线了。

要了解有关 defaultdict 的更多信息，请使用此链接： Click here

Answer 7

这仅适用于宽度和高度相等的矩阵。 但它也不依赖任何第三方。

matrix = [[11, 2, 4],[4, 5, 6],[10, 8, -12]]

# only works for diagnoals of equal width and height
def forward_diagonal(matrix):
    if not isinstance(matrix, list):
        raise TypeError("Must be of type list")

    results = []
    x = 0
    for k, row in enumerate(matrix):
        # next diag is (x + 1, y + 1)
        for i, elm in enumerate(row):

            if i == 0 and k == 0:
                results.append(elm)
                break
            if (x + 1 == i):
                results.append(elm)
                x = i
                break
    return results

print 'forward diagnoals', forward_diagonal(matrix)

Answer 8

基于 Nemo 上述回答的代码：

def print_diagonals(matrix):
    n = len(matrix)
    diagonals_1 = []  # lower-left-to-upper-right diagonals
    diagonals_2 = []  # upper-left-to-lower-right diagonals
    for p in range(2*n-1):
        diagonals_1.append([matrix[p-q][q] for q in range(max(0, p - n + 1), min(p, n - 1) + 1)])
        diagonals_2.append([matrix[n-p+q-1][q] for q in range(max(0, p - n + 1), min(p, n - 1) + 1)])
    print("lower-left-to-upper-right diagonals: ", diagonals_1)
    print("upper-left-to-lower-right diagonals: ", diagonals_2)


print_diagonals([
    [1, 2, 1, 1],
    [1, 1, 4, 1],
    [1, 3, 1, 6],
    [1, 7, 2, 5],
])

lower-left-to-upper-right diagonals:  [[1], [1, 2], [1, 1, 1], [1, 3, 4, 1], [7, 1, 1], [2, 6], [5]]
upper-left-to-lower-right diagonals:  [[1], [1, 7], [1, 3, 2], [1, 1, 1, 5], [2, 4, 6], [1, 1], [1]]

Answer 9

Python 方法

对于纯 Python 实现，我建议在 1D 中工作。

W, H = len(mat[0]), len(mat) 
idx = range(W-1) + range(W-1, W*H, W)
rng = range(1, W) + range(H, 0, -1)
rng = map(lambda x: x if (x < min(W, H)) else min(W, H), rng)
dia = [[i + (W-1) * m for m in xrange(r)] for i, r in zip(idx, rng)]

这里dia 返回每​​个对角线的索引列表。要检索相应的值：

arr = [e for row in mat for e in row] #Flatten the matrix
for d in dia:
    print [arr[e] for e in d][::-1]

[-2]
[9, 5]
[3, -6, 3]
[-1, 2, 5, 2]
[8, 7, 1]
[-4, 3]
[8]

如果要反方向返回值：

arr2 = [e for row in zip(*mat[::-1]) for e in row] #Flatten and rotate the matrix by 90°
for d in dia[::-1]:
    print [arr2[e] for e in d]

[2]
[3, 1]
[5, 5, 3]
[-2, -6, 7, 8]
[9, 2, -4]
[3, 8]
[-1]

Numpy 方法

tril = [np.flip(np.fliplr(mat).diagonal(n)) for n in xrange(mat.shape[0])][::-1]
trir = [np.flipud(mat).diagonal(n) for n in xrange(1, mat.shape[0])]
dia = tril + trir

[array([-2]),
 array([9, 5]),
 array([ 3, -6,  3]),
 array([-1,  2,  5,  2]),
 array([8, 7, 1]),
 array([-4,  3]),
 array([8])]

Answer 10

试试这个：

import numpy as np
matrix = [[-2,  5,  3,  2],
          [ 9, -6,  5,  1],
          [ 3,  2,  7,  3],
          [-1,  8, -4,  8]]

matrix = np.array(matrix)
matrix = np.flipud(matrix)
a = matrix.shape[0]
list_ = [np.diag(matrix, k=i).tolist() for i in range(-a+1,a)]
print(list_)

输出：

[[-2], [9, 5], [3, -6, 3], [-1, 2, 5, 2], [8, 7, 1], [-4, 3], [8]]

Answer 11

尝试使用字典

mat = [[-2,  5,  3,  2],
      [ 9, -6,  5,  1],
      [ 3,  2,  7,  3],
      [-1,  8, -4,  8]]
dct = dict()
for i in range(len(mat)-1,-len(mat[0]),-1):
    dct[i] = []
for i in range(len(mat)):
    for j in range(len(mat[0])):
       dct[i-j].append(mat[i][j])
print(dct)

输出：

{3: [-1], 2: [3, 8], 1: [9, 2, -4], 0: [-2, -6, 7, 8], -1: [5, 5, 3], -2: [3, 1], -3: [2]}

Answer 12

使用迭代工具

matrix = [[-2,  5,  3,  2],
      [ 9, -6,  5,  1],
      [ 3,  2,  7,  3],
      [-1,  8, -4,  8]]

import itertools as it

def show_diagonals(alist):

    # get row/col lenght
    a = len(alist)

    # creating a fliped matrix
    rlist = []
    for r in alist:
        new = r.copy()
        new.reverse()
        rlist.append(new)

    flatten_list = list(it.chain.from_iterable(alist)) 
    flatten_rlist = list(it.chain.from_iterable(rlist)) 
    b = len(flatten_list)
    first_diag = list(it.islice(flatten_list, 0, b+1, a+1))
    second_diag = list(it.islice(flatten_rlist, 0, b+1, a+1))
    return first_diag, second_diag

a, b = show_diagonals(matrix)

Answer 13

使用一些 numpy-fu 获取主对角线：

import numpy as np  
r = np.arange(36) 
r.resize((6, 6)) 
print(r) 
r = r.reshape(len(r)**2)[::len(r)+1] 
print(r)

打印：

[[ 0  1  2  3  4  5]
 [ 6  7  8  9 10 11]
 [12 13 14 15 16 17]
 [18 19 20 21 22 23]
 [24 25 26 27 28 29]
 [30 31 32 33 34 35]]

[ 0  7 14 21 28 35]

Answer 14

从这里：np.Diagonal

 np.diagonal(matrix)