# 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:

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

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.

### Pre-Multiplication

Let $\mathbf A \mathbf B$ be the product of $\mathbf A$ with $\mathbf B$.

Then $\mathbf B$ is **pre-multiplied** by $\mathbf A$.

### Post-Multiplication

Let $\mathbf A \mathbf B$ be the product of $\mathbf A$ with $\mathbf B$.

Then $\mathbf A$ is **post-multiplied** by $\mathbf B$.

### Using Einstein Summation Convention

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

Then:

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

The index which appears twice in the expressions on the right hand side is the entry $k$, which is the one summated over.

### Conformable Matrices

It needs to be emphasised that matrix product can be defined on $\mathbf A$ and $\mathbf B$ if and only if $\mathbf A$ and $\mathbf B$ are **conformable**.

That is, if the number of rows of one is equal to the number of columns of the other.

## Notation

To denote the **(conventional) 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 the **(conventional) matrix product** as the **Cauchy product** after Augustin Louis Cauchy.

However, the latter has a different definition on $\mathsf{Pr} \infty \mathsf{fWiki}$, and is applied to infinite series.

Further rumours suggest that Jacques Philippe Marie Binet may also have lent his name to this concept.

However, corroboration has proven difficult to obtain.

## Examples

### $2 \times 2$ Real Matrices

Let $\mathbf A = \begin {pmatrix} p & q \\ r & s \end {pmatrix}$ and $\mathbf B = \begin {pmatrix} w & x \\ y & z \end {pmatrix}$ be order $2$ square matrices over the real numbers.

Then the matrix product of $\mathbf A$ with $\mathbf B$ is given by:

- $\mathbf A \mathbf B = \begin {pmatrix} p w + q y & p x + q z \\ r w + s y & r x + s z \end {pmatrix}$

### $3 \times 3$ Matrix-Vector Multiplication Formula

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

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

### $2 \times 3$ by $3 \times 2$ Matrices

Let $\mathbf A = \begin {pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end {pmatrix}$ and $\mathbf B = \begin {pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ b_{31} & b_{32} \end {pmatrix}$ be matrices over the real numbers.

Then the matrix product of $\mathbf A$ with $\mathbf B$ is given by:

- $\mathbf A \mathbf B = \begin {pmatrix} a_{11} b_{11} + a_{12} b_{21} + a_{13} b_{31} & a_{11} b_{12} + a_{12} b_{22} + a_{13} b_{32} \\ a_{21} b_{11} + a_{22} b_{21} + a_{23} b_{31} & a_{21} b_{12} + a_{22} b_{22} + a_{23} b_{32} \end {pmatrix}$

### Cayley's Motivation

Let there be $3$ Cartesian coordinate systems:

- $\tuple {x, y}$, $\tuple {x', y'}$, $\tuple {x' ', y' '}$

Let them be connected by:

- $\begin {cases} x' = x + y \\ y' = x - y \end {cases}$

and:

- $\begin {cases} x' ' = -x' - y' \\ y' ' = -x' + y' \end {cases}$

The relationship between $\tuple {x, y}$ and $\tuple {x' ', y' '}$ is given by:

- $\begin {cases} x'' = -x' - y' = -\paren {x + y} - \paren {x - y} = -2 x \\ y'' = -x' + y' = -\paren {x + y} + \paren {x - y} = -2 y \end {cases}$

Arthur Cayley devised the compact notation that expressed the changes of coordinate systems by arranging the coefficients in an array:

- $\begin {pmatrix} 1 & 1 \\ 1 & -1 \end {pmatrix} \begin {pmatrix} -1 & -1 \\ -1 & 1 \end {pmatrix} = \begin {pmatrix} -2 & 0 \\ 0 & -2 \end {pmatrix}$

As such, he can be considered as having invented matrix multiplication.

## Also see

- Results about
**(conventional) matrix multiplication**can be found**here**.

## Historical Note

This mathematical process defined by the (conventional) matrix product was first introduced by Jacques Philippe Marie Binet.

## Linguistic Note

Some older sources use the term **matric multiplication** instead of **matrix 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.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 29$. Matrices - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $2$: Some examples of rings: Ring Example $7$ - 1998: Richard Kaye and Robert Wilson:
*Linear Algebra*... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.2$ Addition and multiplication of matrices - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**matrix**(*plural***matrices**) - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.2$: Functions on vectors: $\S 2.2.4$: Multiplication and inverses - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**matrix**(*plural***matrices**)