The orthogonal projection (or simply ``projection'') of
onto
is defined by

The complex scalar
is called the
coefficient of projection. When projecting onto a unit
length vector , the coefficient of projection is simply the inner
product of with .

Motivation: The basic idea of orthogonal projection of onto
is to ``drop a perpendicular'' from onto to define a new
vector along which we call the ``projection'' of onto .
This is illustrated for in Fig.5.9 for and
, in which case

Figure 5.9:
Projection of
onto
in 2D space.

Derivation: (1) Since any projection onto must lie along the
line collinear with , write the projection as
. (2) Since by definition the projection error
is orthogonal to , we must have

Thus,

See §I.3.3 for illustration of orthogonal projection in matlab.