Projection

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

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

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

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

$${\bf P}_{\underline{x}}(\underline{y}) \isdef \frac{\left<\underl... ...ne{x} = \frac{11}{17} \underline{x}= \left[\frac{44}{17},\frac{11}{17}\right].$$

Figure: Projection of $\underline{y}$ onto $\underline{x}$ in 2D space.

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

$$\begin{eqnarray*} (\underline{y}-\alpha\underline{x}) & \perp & \underline{x}\\ ... ...<\underline{y},\underline{x}\right>}{\Vert\underline{x}\Vert^2}. \end{eqnarray*}$$

Thus,

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