nLab Hadamard product

Definition

Of matrices

The Hadamard product or Schur product or element-wise product of two matrices $A$ and $B$ in a commutative ring $R$ is a matrix $A \odot B$ whose elements $c_{i, j}$ are the product $a_{i, j} \cdot b_{i, j}$ of the respective elements $a_{i, j}$ in the matrix $A$ and $b_{i, j}$ in the matrix $B$.

Of power series

The Hadamard product of two power series $p = \sum_{i = 0}^\infty a_i x^i$ and $q = \sum_{i = 0}^\infty b_i x^i$ with coefficients from a commutative ring $R$ is the power series defined by

$$p \odot q \coloneqq \sum_{i = 0}^\infty a_i b_i x^i$$

More generally

Let $R$ be a commutative ring, and let $[n]$ denote the finite set with $n$ elements. The $m$ by $n$ matrix algebra $M_{m, n}(R)$ is isomorphic to the function algebra $[m n] \to R$, where the Hadamard product in $M_{m, n}(R)$ corresponds to pointwise multiplication in $[m n] \to R$. Similarly, the power series ring $R[[x]]$ is isomorphic to the function algebra $\mathbb{N} \to R$, where the Hadamard product in $R[[x]]$ corresponds to pointwise multiplication in $\mathbb{N} \to R$.

Thus, there is a generalization of the definition of Hadamard product from matrix algebras and power series rings to any commutative algebra with a isomorphism of commutative algebras to a function algebra:

Let $S$ be a set, let $R$ be a commutative ring, and let $A$ be a commutative $R$-algebra with an isomorphism of commutative $R$-algebras $i:A \cong (S \to R)$. Then the Hadamard product in $A$ is a binary operation $(X, Y) \mapsto X \odot Y:A \times A \to A$ defined as

$$X \odot Y \coloneqq i^{-1}(x \mapsto i(X)(x) \cdot i(Y)(x))$$

Related concept

• matrix

• power series

References

• Wikipedia, Hadamard product (matrices)

• Wikipeida, Generating function transformation#Hadamard products and diagonal generating functions