# Extract upper or lower triangular part of a numpy matrix

## Question:

I have a matrix `A`

and I want 2 matrices `U`

and `L`

such that `U`

contains the upper triangular elements of A (all elements above and not including diagonal) and similarly for `L`

(all elements below and not including diagonal). Is there a `numpy`

method to do this?

e.g

```
A = array([[ 4., 9., -3.],
[ 2., 4., -2.],
[-2., -3., 7.]])
U = array([[ 0., 9., -3.],
[ 0., 0., -2.],
[ 0., 0., 0.]])
L = array([[ 0., 0., 0.],
[ 2., 0., 0.],
[-2., -3., 0.]])
```

## Answers:

Try `numpy.triu`

(triangle-upper) and `numpy.tril`

(triangle-lower).

Code example:

```
np.triu([[1,2,3],[4,5,6],[7,8,9],[10,11,12]])
array([[ 1, 2, 3],
[ 4, 5, 6],
[ 0, 8, 9],
[ 0, 0, 12]])
```

Use the Array Creation Routines of numpy.triu and numpy.tril to return a copy of a matrix with the elements above or below the k-th diagonal zeroed.

```
>>> a = np.array([[1,2,3],[4,5,6],[7,8,9]])
>>> a
array([[1, 2, 3],
[4, 5, 6],
[7, 8, 9]])
>>> tri_upper_diag = np.triu(a, k=0)
>>> tri_upper_diag
array([[1, 2, 3],
[0, 5, 6],
[0, 0, 9]])
>>> tri_upper_no_diag = np.triu(a, k=1)
>>> tri_upper_no_diag
array([[0, 2, 3],
[0, 0, 6],
[0, 0, 0]])
>>> tri_lower_diag = np.tril(a, k=0)
>>> tri_lower_diag
array([[1, 0, 0],
[4, 5, 0],
[7, 8, 9]])
>>> tri_lower_no_diag = np.tril(a, k=-1)
>>> tri_lower_no_diag
array([[0, 0, 0],
[4, 0, 0],
[7, 8, 0]])
```

To **extract the upper triangle values** to a flat vector,

you can do something like the following:

```
import numpy as np
a = np.array([[1,2,3],[4,5,6],[7,8,9]])
print(a)
#array([[1, 2, 3],
# [4, 5, 6],
# [7, 8, 9]])
a[np.triu_indices(3)]
#or
list(a[np.triu_indices(3)])
#array([1, 2, 3, 5, 6, 9])
```

Similarly, for the **lower triangle**, use `np.tril`

.

*IMPORTANT*

If you want to extract the values that are **above the diagonal** (or **below**) then use the **k** argument. This is usually used when the matrix is symmetric.

```
import numpy as np
a = np.array([[1,2,3],[4,5,6],[7,8,9]])
#array([[1, 2, 3],
# [4, 5, 6],
# [7, 8, 9]])
a[np.triu_indices(3, k = 1)]
# this returns the following
array([2, 3, 6])
```

**EDIT (on 11.11.2019):**

To put back the extracted vector into a 2D symmetric array, one can follow my answer here: https://stackoverflow.com/a/58806626/5025009