Update README.md

Signed-off-by: David Rotermund <54365609+davrot@users.noreply.github.com>

numpy/einstein/README.md

@@ -10,6 +10,77 @@
 
 Questions to [David Rotermund](mailto:davrot@uni-bremen.de)
 
+## Examples
+
+$$\sum_i a_{i,i}$$
+
+```python
+import numpy as np
+
+a = np.arange(0, 25).reshape(5, 5)
+print(a)
+print(a.shape)  # -> (5, 5)
+
+print(np.einsum("ii", a)) # -> 60
+```
+
+Output:
+
+```python
+[[ 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]]
+```
+
+$$b(i) = a_{i,i}$$
+
+```python
+import numpy as np
+
+a = np.arange(0, 25).reshape(5, 5)
+print(np.einsum("ii->i", a))  # -> [ 0  6 12 18 24]
+```
+
+$$b(i) = \sum_j a_{i,j}$$
+
+```python
+import numpy as np
+
+a = np.arange(0, 25).reshape(5, 5)
+print(np.einsum("ij->i", a))  # -> [ 10  35  60  85 110]
+print(np.einsum("...j->...", a))  # -> [ 10  35  60  85 110]
+```
+
+$$b(j,i) = a_{i,j}$$
+
+```python
+import numpy as np
+
+a = np.arange(0, 25).reshape(5, 5)
+print(a)
+print()
+print(np.einsum("ij->ji", a))
+```
+
+Output:
+
+```python
+[[ 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]]
+
+[[ 0  5 10 15 20]
+ [ 1  6 11 16 21]
+ [ 2  7 12 17 22]
+ [ 3  8 13 18 23]
+ [ 4  9 14 19 24]]
+```
+
+
 
 ## [numpy.einsum](https://numpy.org/doc/stable/reference/generated/numpy.einsum.html)