componentwise

Componentwise is a term used in mathematics, computer science, and related fields to describe operations or comparisons performed on corresponding components of structured data such as vectors or matrices, rather than on the structure as a whole. For vectors x = (x1, ..., xn) and y = (y1, ..., yn), many standard operations are defined componentwise: the sum is (x1 + y1, ..., xn + yn), and the Hadamard (elementwise) product is (x1 y1, ..., xn yn). A function applied componentwise to a vector applies the same function to each entry: f(x) = (f(x1), ..., f(xn)).

The concept extends to inequalities through the componentwise order: x ≤ y means xi ≤ yi for all i. This defines a partial order on R^n and yields a lattice structure where the componentwise supremum and infimum are given by coordinate-wise maxima and minima.

Componentwise operations contrast with matrix multiplication, where the result entry i,j depends on a sum over products of row and column elements. In numerical computing, componentwise operations are standard in array programming languages such as NumPy, MATLAB, and R, where operations are applied elementwise by default.

The term is also used in contexts such as componentwise function monotonicity and in discussing convergence and stability in systems of inequalities, where properties hold for each component individually.