Definition:Hadamard Product

Definition

Let $\struct {S, \cdot}$ be an algebraic structure.

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

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

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:

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

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.

This restriction applies to the operation of matrix entrywise addition, which can be considered as a specific application of the Hadamard product.

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, which is the specific application of the Hadamard product to the ring addition operation of a matrix space whose underlying structure is a ring.

• 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