Easy Framework

Matrix Multiplication

Multiply two matrices by taking a dot product between each row of A and each column of B.

Your task

Implement matmul(A, B) using the framework's built-in matrix multiplication. A has shape (m, k) and B has shape (k, n). Both are floating-point tensors with the same dtype and device. Return shape (m, n), preserving dtype and device and leaving the inputs unchanged.

output[i, j] = sum(A[i, t] * B[t, j] for t from 0 to k-1)

For A=[[1, 2], [3, 4]] and B=[[5, 6], [7, 8]], return [[19, 22], [43, 50]]. The top-left value is 1*5 + 2*7 = 19: A's first row is paired with B's first column.

The shared inner dimension k is summed over. The remaining dimensions m and n become the output rows and columns. Matrices need not be square: (2, 3) multiplied by (3, 4) produces (2, 4).

Use A @ B, torch.matmul(A, B) or jnp.matmul(A, B). A * B is elementwise multiplication and does not compute these row-column sums. Keep the input order; B @ A may have a different result or incompatible shapes. Keep a matrix result even when m or n is 1.

Understand the ideas behind this exercise. The Problem tab contains the exact implementation requirements.

Multiply rows by columns

For A with shape (M, K) and B with shape (K, N), the matrix product has shape (M, N). Each output entry is a dot product of one row of A and one column of B:

$$C_{ij}=\sum_{k=1}^{K}A_{ik}B_{kj}.$$

For example, multiplying row [1, 2] by column [3, 4] gives 1 * 3 + 2 * 4 = 11. Elementwise multiplication would instead give [3, 8]; it does not include the sum.

If examples are rows of X, with shape (B, D), and weights have shape (D, H), then XW produces H features for each of the B examples.

Keep batch axes separate from matrix axes

For batched matrix multiplication, the last two axes hold each matrix. Leading axes identify batches of matrices. A general matrix-multiply operation may broadcast compatible batch axes. A stricter batched operation may require them to match exactly. Use the operation specified by the exercise.

Einstein notation, often used with einsum, names the axes explicitly:

bij,bjk->bik multiplies corresponding matrices in batch b and sums over j.
bi,bj->bij forms an outer product for each example: output (b, i, j) is the product of input (b, i) and input (b, j).

An axis is summed out when its label is absent from the explicit output. A label that appears in both inputs is not necessarily summed: b stays in both outputs above.

Choose a multiplication order with small intermediate results

In exact arithmetic, (AB)C = A(BC), but the two orders can have very different costs. If the shapes are (1000, 10), (10, 10), and (10, 1), forming BC first needs a (10, 1) intermediate. Forming AB first needs a (1000, 10) intermediate and many more operations.

Floating-point rounding can also differ between orders, so do not expect every bit of the two results to match.

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Matrix Multiplication

Your task

Multiply rows by columns

Keep batch axes separate from matrix axes

Choose a multiplication order with small intermediate results

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