Definition:Matrix Product (Conventional)

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Definition

Let $\left({R, +, \circ}\right)$ be a ring.

Let $\mathbf A = \left[{a}\right]_{m n}$ be an $m \times n$ matrix over $R$.

Let $\mathbf B = \left[{b}\right]_{n p}$ 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 $\mathbf A \mathbf B = \mathbf C = \left[{c}\right]_{m p}$.


Then:

$\displaystyle \forall i \in \left[{1 \,.\,.\, m}\right], j \in \left[{1 \,.\,.\, p}\right]: c_{i j} = \sum_{k \mathop = 1}^n a_{i k} \circ b_{k j}$


Thus $\left[{c}\right]_{m p}$ is the $m \times p$ matrix where each element $c_{i j}$ is built by forming the product of each element in the $i$'th row of $\mathbf A$ with the corresponding element 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.


Using Summation Convention

The matrix product of $\mathbf A$ and $\mathbf B$ can be expressed using the summation convention as:

Then:

$c_{i j} := a_{i k} \circ b_{k j}$


The index which appears twice in the expressions on the RHS is the element $k$, which is the one summated over.


Examples

The $3 \times 3$ matrix-vector multiplication formula is an instance of the Matrix Product operation:

$\mathbf A \mathbf v = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} a_{11} x + a_{12} y + a_{13} z \\ a_{21} x + a_{22} y + a_{23} z \\ a_{31} x + a_{32} y + a_{33} z \\ \end{bmatrix}$


Linguistic Note

Some older sources use the term matric multiplication; strictly speaking it is more correct, as matric is the adjective formed from the noun matrix, but it is a little old-fashioned and is rarely found nowadays.


Historical Note

This mathematical process was first introduced by Jacques Philippe Marie Binet.


Beware

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


Also see


Sources