Einsum¶
If we repeated letters we want those axes multiplied together
If we omit a letter from the output it means that values along that axis will be summed over For 1D arrays
| Call signature | NumPy equivalent | Description |
|---|---|---|
('i', A) |
A |
returns a view of A |
('i->', A) |
sum(A) |
sums the values of A |
('i,i->i', A, B) |
A * B |
element-wise multiplication of A and B |
('i,i', A, B) |
inner(A, B) |
inner product of A and B |
('i,j->ij', A, B) |
outer(A, B) |
outer product of A and B |
For matrices
| Call signature | NumPy equivalent | Description |
|---|---|---|
('ij', A) |
A |
returns a view of A |
('ji', A) |
A.T |
view transpose of A |
('ii->i', A) |
diag(A) |
view main diagonal of A |
('ii', A) |
trace(A) |
sums main diagonal of A |
('ij->', A) |
sum(A) |
sums the values of A |
('ij->j', A) |
sum(A, axis=0) |
sum down the columns of A (across rows) |
('ij->i', A) |
sum(A, axis=1) |
sum horizontally along the rows of A |
('ij,ij->ij', A, B) |
A * B |
element-wise multiplication of A and B |
('ij,ji->ij', A, B) |
A * B.T |
element-wise multiplication of A and B.T |
('ij,jk', A, B) |
dot(A, B) |
matrix multiplication of A and B |
('ij,kj->ik', A, B) |
inner(A, B) |
inner product of A and B |
('ij,kj->ikj', A, B) |
A[:, None] * B |
each row of A multiplied by B |
('ij,kl->ijkl', A, B) |
A[:, :, None, None] * B |
each value of A multiplied by B |
Ellipse syntax .... This allows to nod index dimensions:
np.einsum('...ij,ji->...', a, b )
This multiplies the last wo axes of a with an 2d array b.