Definition:Hadamard Product

Definition

Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{m n}$ be $m \times n$ matrices over a ring $\struct {R, +, \times}$.

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:

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

Defined Operation

It needs to be noted that the operation of Hadamard product is defined only when both matrices have the same number of rows and the same number of columns.

Also known as

The Hadamard product is also known as:

the (matrix) entrywise product

the Schur product, for Issai Schur.

Also see

• Definition:Matrix Entrywise Addition

• Results about Hadamard product can be found here.

Source of Name

This entry was named for Jacques Salomon Hadamard.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hadamard product