Definition:Matrix Product (Conventional)

Definition
Let $\struct {R, +, \circ}$ be a ring.

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

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

Then the matrix product of $\mathbf A$ and $\mathbf B$ is written $\mathbf A \mathbf B$ and is defined as follows.

Let $\mathbf A \mathbf B = \mathbf C = \sqbrk c_{m p}$.

Then:
 * $\displaystyle \forall i \in \closedint 1 m, j \in \closedint 1 p: c_{i j} = \sum_{k \mathop = 1}^n a_{i k} \circ b_{k j}$

Thus $\sqbrk c_{m p}$ is the $m \times p$ matrix where each entry $c_{i j}$ is built by forming the (ring) product of each entry in the $i$'th row of $\mathbf A$ with the corresponding entry in the $j$'th column of $\mathbf B$ and adding up all those products.

This operation is called matrix multiplication, and $\mathbf C$ is the matrix product of $\mathbf A$ with $\mathbf B$.

It follows that matrix multiplication is defined whenever the first matrix has the same number of columns as the second matrix has rows.

Notation
To denote the matrix product of $\mathbf A$ with $\mathbf B$, the juxtaposition notation is always used:
 * $\mathbf C = \mathbf A \mathbf B$

We do not use $\mathbf A \times \mathbf B$ or $\mathbf A \cdot \mathbf B$ in this context, because they have specialised meanings.

Also known as
It is believed that some sources refer to this as the Cauchy product after.

Further rumours suggest that may also have lent his name to this concept.

However, corroboration has proven difficult to obtain.

Also see

 * Definition:Matrix Product