Matrices and matrix multiplication

A matrix is a grid of numbers with rows and columns. A 2×3 matrix has 2 rows and 3 columns. In Python it is a list of rows: [[1, 2, 3], [4, 5, 6]].

Multiplying matrices

Entry (i, j) of A·B is row i of A dotted with column j of B. So the rows of A must be as long as the columns of B:

(m×n) times (n×p) gives m×p

A 2×3 matrix times a 3×4 matrix is 2×4. A 2×3 times a 2×3 cannot be multiplied at all.

A = [[1, 2],
     [3, 4]]
B = [[0, 1],
     [1, 0]]

def matmul(X, Y):
    return [[sum(X[i][k] * Y[k][j] for k in range(len(Y))) for j in range(len(Y[0]))]
            for i in range(len(X))]

print("A·B =", matmul(A, B))
print("B·A =", matmul(B, A))
print("A times the column (1, 1) =", matmul(A, [[1], [1]]))

Order matters

In the example, B swaps things around. A·B swaps the columns of A, while B·A swaps its rows: two different answers. For matrices, A·B and B·A are usually different.

The identity

The identity matrix I has 1s on the diagonal and 0s elsewhere. A·I = I·A = A, just as 1 × x = x.

Why it matters

A matrix times a vector transforms the vector: rotating a game character, applying a filter to an image, or computing one layer of a neural network. Graphics cards are built to do millions of these multiplications per second.

Common mistakes

• Multiplying entry by entry (A[i][j] × B[i][j]). That is a different operation, not matrix multiplication.
• Checking the wrong sizes. The inner sizes must match: (2×3)(3×4).

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