# Definition:Matrix Scalar Product

## Contents

## Definition

Let $\GF$ denote one of the standard number systems.

Let $\map \MM {m, n}$ be the $m \times n$ matrix space over $\GF$.

Let $\mathbf A = \sqbrk a_{m n} \in \map \MM {m, n}$.

Let $\lambda \in \GF$ be any element of $\Bbb F$.

The operation of **scalar multiplication of $\mathbf A$ by $\lambda$** is defined as follows.

Let $\lambda \mathbf A = \mathbf C$.

Then:

- $\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j} = \lambda a_{i j}$

$\lambda \mathbf A$ is the **scalar product of $\lambda$ and $\mathbf A$**.

Thus $\mathbf C = \sqbrk c_{m n}$ is the $m \times n$ matrix composed of the product of $\lambda$ with the corresponding elements of $\mathbf A$.

### Ring

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

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

Let $\lambda \in R$ be any element of $R$.

The **scalar product of $\lambda$ and $\mathbf A$** is defined as follows.

Let $\lambda \circ \mathbf A = \mathbf C$.

Then:

- $\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j} = \lambda \circ a_{i j}$

Thus $\sqbrk c_{m n}$ is the $m \times n$ matrix composed of the product of $\lambda$ with the corresponding elements of $\mathbf A$.

### Scalar

The element $\lambda$ of the underlying structure of $\map \MM {m, n}$ is known as a **scalar**.

## Also see

- Results about
**matrix scalar product**can be found here.

## Sources

- 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