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

denotes a vector of

complex elements, and

denotes a complex scalar,
then

denotes the
scalar multiplication of

by

. 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.
Next Section: Linear Combination of VectorsPrevious Section: Vector Subtraction