Scalar Multiplication

A scalar is any constant value used as a scale factor applied to a vector. Mathematically, all of our scalars will be either real or complex numbers.^5.3 For example, if $\underline{x}\in{\bf C}^N$ denotes a vector of $N$ complex elements, and $\alpha\in{\bf C}$ denotes a complex scalar, then

$$\alpha\, \underline{x}\isdef (\alpha\,x_1, \alpha\,x_2, \ldots, \alpha\,x_N)$$

denotes the scalar multiplication of $\underline{x}$ by $\alpha$. Thus, multiplication of a vector by a scalar is done in the obvious way, which is to multiply each coordinate of the vector by the scalar.

In signal processing, we think of scalar multiplication as applying some constant scale factor to a signal, i.e., multiplying each sample of the signal by the same constant number. For example, a 6 dB boost can be carried out by multiplying each sample of a signal by 2, in which case 2 is the scalar. When the scalar magnitude is greater than one, it is often called a gain factor, and when it is less than one, an attenuation.