Pairwise cosine-similarity matrix

Background

Cosine similarity measures the angle between two vectors, ignoring their magnitudes: $+1$ for the same direction, $0$ for orthogonal, $-1$ for opposite. It is the similarity for embeddings — semantic search, RAG retrieval, recommendation, and clustering all rank candidates by cosine similarity to a query.

Problem statement

Implement cosine_similarity_matrix(A, B) returning the matrix S of cosine similarities between every row of A and every row of B:

$$S_{ij} = \frac{A_i \cdot B_j}{\lVert A_i\rVert\,\lVert B_j\rVert}$$

A zero vector has no direction, so its similarity to anything is defined as $0$.

Input

• A — np.ndarray of shape (n, d).
• B — np.ndarray of shape (m, d).

Output

Returns an np.ndarray of shape (n, m) where S[i, j] is the cosine similarity of A[i] and B[j].

Examples

Example 1

Input:  A = [[1, 0], [0, 1]], B = [[1, 1], [1, 0]]
Output: [[0.7071, 1.0], [0.7071, 0.0]]

Explanation: $[1,0]$ vs $[1,1]$ is $\cos 45^\circ = 0.7071$ and vs $[1,0]$ is $1$; $[0,1]$ vs $[1,1]$ is $0.7071$ and vs $[1,0]$ is $0$ (orthogonal).

Constraints

• Normalise each row to unit length, then take the dot products (equivalently divide each $A_i \cdot B_j$ by the two norms).
• A zero-norm row contributes similarity $0$ — do not divide by zero.
• Output shape (n, m); values in $[-1, 1]$; tests compare with atol=1e-6.

Notes

• Pre-normalising the rows turns the whole computation into one matrix product $\hat{A}\hat{B}^\top$ — the efficient way to score a query against a large corpus of embeddings.
• Cosine ignores magnitude, so it is robust to document length and embedding scale; Euclidean distance on normalised vectors is monotonic with cosine.