Definition:Matrix Scalar Product

Definition
Let $\Bbb F$ denote one of the standard number systems.

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

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

Let $\lambda \in R$ 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 $\mathbf A$.

Also see

 * Definition:Matrix Product
 * Definition:Scalar (Matrix Theory)