How to compute outer product using Python

An outer product, also known as a vector matrix product, is a kind of vector multiplication that outputs a matrix. Suppose we have two column vectors $X$and $Y$of dimensions $m \times 1$ and $n \times 1$respectively.

The outer product of these vectors will be a matrix $M$of shape $m \times n$.

The outer product is represented by the symbol $\otimes$. Taking the outer product of two vectors is equivalent to multiplying the first vector with the transpose of the second.

The first element of the column vector $X$ will multiply with all the elements of row vector $Y$, then the second element of column vector $X$ will multiply with all the elements of row vector $Y$, and so on until the entire $m \times n$ matrix has been populated.

How to calculate the outer product in Python?

To calculate the outer product of two vectors in Python, we can use a function provided by the NumPy library named outer(). To check how it works, let's consider the following two simple column vectors.

The outer product of the two will be:

The code below shows how this outer product can be calculated using the outer() function.

Code explanation

• Line 1: Import the numpy library.

• Lines 3–4: Define two lists that will be treated as vectors.

• Line 6: Call NumPy’s outer() function to calculate the outer product of the two vectors.

• Line 8: Display the computed matrix on the screen.

Knowledge test

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Conclusion

By employing the numpy.outer() function, users can compute the outer product of two vectors, facilitating various applications in mathematics, statistics, and data analysis. Through this article, we have explored the concept of the outer product, its mathematical representation, and its implementation in Python using NumPy, providing readers with a comprehensive understanding of this fundamental operation.