Fieldwise

Fieldwise is an informal term used in mathematics and related disciplines to describe operations that are applied independently to each entry of a data object whose elements come from a mathematical field. A field is a set equipped with addition and multiplication satisfying the usual axioms (such as the real numbers, complex numbers, or finite fields). Fieldwise operations are typically described as entrywise or coordinatewise, emphasizing that each entry is transformed or combined without mixing entries across positions.

In linear algebra, fieldwise operations on matrices over a field F are defined entry by entry. For example, the fieldwise sum of two matrices A and B of the same size is a matrix C with c_ij = a_ij + b_ij. The fieldwise product, often called the Hadamard product, is the matrix D with d_ij = a_ij b_ij. More generally, applying a function f to a matrix entrywise yields f(A) = [f(a_ij)], provided f is defined on the elements of F. This approach is distinct from standard matrix multiplication, which combines entries across rows and columns.

In data processing and scientific computing, fieldwise operations describe columnwise or fieldwise processing of data: normalizing each field (column) separately, or applying a transformation to each field of a structured dataset. The term highlights per-entry operations and is often contrasted with row-wise or matrix-wide computations.

Notes: fieldwise is less formal than terms like entrywise or componentwise and may appear primarily in applied or informal contexts. When precision is important, it is common to specify entrywise, Hadamard, or coordinatewise to avoid ambiguity.