# Definition:Hadamard Product

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## 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**