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:


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

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.