Скачать или смотреть How to Multiply Numpy Arrays Without Intermediate Arrays Using np.einsum and Outer Products

Learn how to multiply 1D numpy arrays of incompatible shapes efficiently without creating intermediate arrays, using np.einsum and outer products.
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This video is based on the question https://stackoverflow.com/q/79537428/ asked by the user 'Paul Jurczak' ( https://stackoverflow.com/u/219153/ ) and on the answer https://stackoverflow.com/a/79537727/ provided by the user 'hpaulj' ( https://stackoverflow.com/u/901925/ ) at 'Stack Overflow' website. Thanks to these great users and Stackexchange community for their contributions.

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The Problem: Multiplying Numpy Arrays With Different Shapes

When you have two 1D numpy arrays, such as:

[[See Video to Reveal this Text or Code Snippet]]

Attempting simple element-wise multiplication like a * c results in a ValueError due to shape mismatch—numpy cannot broadcast (6,) and (2,) directly because they differ in size and dimensions.

Why Does This Error Occur?

Numpy's broadcasting rules require that:

Arrays have the same dimensions, or

One of the arrays must be broadcastable to the other's shape.

In this case, (6,) and (2,) are incompatible vectors. Reshaping or expanding c is one approach, but it can still create intermediary arrays.

Desired Goal

To multiply both arrays to produce a matrix where each row of a is multiplied by each element of c, resulting in a 2D array with shape (6, 2), without creating extra large intermediate arrays explicitly.

Solutions

1. Using np.einsum for Explicit Outer Product

np.einsum lets you define summations and multiplications on indices explicitly, often more memory-efficient and expressive:

[[See Video to Reveal this Text or Code Snippet]]

This computes the outer product, producing a (6, 2) array where each element of a multiplies each element of c.

2. Using np.outer or np.multiply.outer

Numpy provides convenient functions:

[[See Video to Reveal this Text or Code Snippet]]

Both produce the same (6,2) shaped output.

Example

[[See Video to Reveal this Text or Code Snippet]]

Output:

[[See Video to Reveal this Text or Code Snippet]]

Summary

Simple element-wise multiplication fails on incompatible 1D array shapes.

Use np.einsum('i,j->ij', a, c) to create the outer product directly without broadcasting errors.

Alternatively, use np.outer(a, c) or np.multiply.outer(a, c).

These methods efficiently produce a 2D array without explicit intermediate reshaping.

This approach is clean, concise, and uses numpy’s built-in optimized functions suitable for modern array operations.