(Redirected from Definition:Schur Product)

## Definition

Let $\struct {R, +_R, \times_R}$ be a ring.

Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over $R$.

Let $\mathbf B = \sqbrk b_{m n}$ be an $m \times n$ matrix over $R$.

The Hadamard product of $\mathbf A$ and $\mathbf B$ is written $\mathbf A \circ \mathbf B$ and is defined as follows:

$\mathbf A \circ \mathbf B := \mathbf C = \sqbrk c_{m n}$

where:

$\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j} = a_{i j} \times_R b_{i j}$

## Also known as

Also called the entrywise product or the Schur product.

## Source of Name

This entry was named for Jacques Salomon Hadamard.