elementprodukt

Elementprodukt is a term used in mathematics and related fields to describe the element-wise product of two sequences or arrays of the same size. In this operation, each entry of the result is the product of the corresponding entries from the operands. It is widely known in linear algebra as the Hadamard product (also called the Schur product) and is applicable to vectors as well as matrices.

Formally, if a = (a1, ..., an) and b = (b1, ..., bn) are vectors, their elementprodukt is a ∘ b = (a1 b1, ..., an bn). For matrices of the same dimensions, the operation is performed entrywise, so A ∘ B is defined by (A ∘ B)ij = Aij Bij. The symbol ∘ is commonly used, though other notations such as ⊙ or elementwise multiplication appear in some texts and software environments.

Key properties include that the elementprodukt is commutative (a ∘ b = b ∘ a) and associative with respect to the same-sized operands, and it distributes over addition (a ∘ (b + c) = a ∘ b + a ∘ c). An identity element exists, namely the vector or matrix of ones, such that a ∘ 1 = a. When components are nonzero, a componentwise inverse can be defined as a^{-1} = (1/a1, ..., 1/an), giving a ∘ a^{-1} = 1.

Applications span data processing, image and signal processing, and machine learning, where elementwise operations are used for scaling, masking, activation functions, and feature-wise computations. The Hadamard product is distinct from the dot product, which combines components via summation.