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
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.