Scalar Multiplication

Definition: Scalar Multiplication

The scalar multiplication of a matrix $A\in F^{m\times n}$ with a field element $\lambda \in F$ is done component-wise through the multiplication operation of $F$ and also results in an $m\times n$-matrix:

$$\lambda \cdot \left[\begin{array}{ccc}{a}_{11}& \cdots & a_{1n}\\ ⋮& \ddots & ⋮\\ a_{m1}& \cdots & a_{mn}\end{array}\right]\stackrel{\text{def}}{=}\left[\begin{array}{ccc}\lambda \cdot a_{11}& \cdots & \lambda \cdot a_{1n}\\ ⋮& \ddots & ⋮\\ \lambda \cdot a_{m1}& \cdots & \lambda \cdot a_{mn}\end{array}\right]\in F^{m\times n}$$