Matrices

A matrix is a 2-D array of numbers with rows and columns.

Matrices

A matrix is a two-dimensional table of numbers. In ML a dataset is a matrix: each row of X is an example and each column a feature. The weights of a neural network layer are a matrix too.

A = np.array([[1, 2],
              [3, 4],
              [5, 6]])
print(A.shape)   # (3, 2): a 3x2 matrix
print(A[1])      # [3 4]   second row

Like vectors, matrices are added element-wise and multiplied by scalars; the shapes must match.

Broadcasting

NumPy automatically expands arrays of different shapes if they follow the rules. The most common use is subtracting a vector from every row of a matrix:

X = np.array([[1, 10],
              [3, 20],
              [5, 30]])
ort = X.mean(axis=0)   # [ 3. 20.]  shape (2,)
print(X - ort)         # subtract the mean from every row
# [[-2. -10.] [ 0.   0.] [ 2.  10.]]

The rule: starting from the right, each dimension must be equal or 1.

Exercise Centre

Subtract the column means of X from every row using broadcasting and print the result.

Show example solution

import numpy as np

X = np.array([[2, 8], [4, 6], [6, 4]])

print(X - X.mean(axis=0))

Test yourself

Can two matrices of shapes (3, 2) and (2, 3) be added with <code>+</code>?
- Yes, it becomes (3, 3)
- Yes, it becomes (2, 2)
- Yes, always
- No; the shapes don't match
Show answer

No; the shapes don't match — In element-wise operations the shapes must be the same or broadcastable; (3, 2) and (2, 3) aren't.
Subtract the column means from every row.
```
Xc = X - X.____(axis=0)
```
Show answer
```
Xc = X - X.mean(axis=0)
```
Match the shape pairs with the results.

(3, 2) + (2,)

(3, 2)
(3, 2) + (3, 1)

(3, 2) broadcast
(3, 2) + (3,)

Error
(3, 2) * 2

Scalar multiplication

Next lesson: Matrix multiplication

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