Projection

The orthogonal projection (or simply ``projection'') of $y\in\mathbb{C}^N$ onto $x\in\mathbb{C}^N$ is defined by

$$\zbox {{\bf P}_{x}(y) \isdef \frac{\left<y,x\right>}{\Vert x\Vert^2} x.}$$

The complex scalar $\left<y,x\right>/\Vert x\Vert^2$ is called the coefficient of projection. When projecting $y$ onto a unit length vector $x$ , the coefficient of projection is simply the inner product of $y$ with $x$ .

Motivation: The basic idea of orthogonal projection of $y$ onto $x$ is to ``drop a perpendicular'' from $y$ onto $x$ to define a new vector along $x$ which we call the ``projection'' of $y$ onto $x$ . This is illustrated for $N=2$ in Fig.5.9 for $x= [4,1]$ and $y=[2,3]$ , in which case

$${\bf P}_{x}(y) \isdef \frac{\left<y,x\right>}{\Vert x\Vert^2} x = \frac{(2\cdot \overline{4} + 3\cdot \overline{1})}{4^2+1^2} x = \frac{11}{17} x= \left[\frac{44}{17},\frac{11}{17}\right].$$

Figure 5.9: Projection of $y$ onto $x$ in 2D space.

Derivation: (1) Since any projection onto $x$ must lie along the line collinear with $x$ , write the projection as ${\bf P}_{x}(y)=\alpha x$ . (2) Since by definition the projection error $y-{\bf P}_{x}(y)$ is orthogonal to $x$ , we must have

$$\begin{eqnarray*} (y-\alpha x) & \perp & x\\ \;\Leftrightarrow\;\left<y-\alpha x,x\right> &=& 0 \\ \;\Leftrightarrow\;\left<y,x\right> &=& \alpha\left<x,x\right> \\ \;\Leftrightarrow\;\alpha &=& \frac{\left<y,x\right>}{\left<x,x\right>} = \frac{\left<y,x\right>}{\Vert x\Vert^2}. \end{eqnarray*}$$

Thus,

$${\bf P}_{x}(y) = \frac{\left<y,x\right>}{\Vert x\Vert^2} x.$$

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