elementweisen

Elementweisen, or elementwise, is a term used in mathematics and computing to describe operations performed on corresponding elements of arrays. An operation is applied independently to each position, producing an output array of the same shape as the inputs. This contrasts with matrix or tensor multiplications that combine elements across positions.

Common elementwise operations include addition, subtraction, multiplication, and division. For vectors a and b of equal length, the elementwise sum is a + b with (a + b)i = ai + bi, and the elementwise product is a ∘ b with (a ∘ b)i = ai · bi. The Hadamard product is the standard name for the elementwise multiplication of matrices of the same dimensions.

In programming languages, elementwise operations are usually distinguished from general matrix operations. For example, MATLAB uses A .* B to perform elementwise multiplication, while A * B computes the matrix product. In Python with NumPy, standard arithmetic operators on arrays perform elementwise operations, and broadcasting allows these operations to apply to arrays of different shapes according to defined rules. Other languages like R and Julia provide similar elementwise semantics, often with concise syntax such as dot operators or built-in broadcasting.

Applications of elementwise computation are widespread in data processing, scientific computing, image and signal processing, and machine learning. The concept underpins vectorization techniques that improve performance by avoiding explicit loops and leveraging optimized, parallel hardware.