Projection

As discussed in §5.9.9, the orthogonal projection of $y\in{\bf C}^N$ onto $x\in{\bf C}^N$ is defined by

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

In matlab, the projection of the length-N column-vector y onto the length-N column-vector x may therefore be computed as follows:

yx = (x' * y) * (x' * x)^(-1) * x

More generally, a length-N column-vector y can be projected onto the $M$-dimensional subspace spanned by the columns of the N $\times$ M matrix X:

yX = X * (X' * X)^(-1) * X' * y

Orthogonal projection, like any finite-dimensional linear operator, can be represented by a matrix. In this case, the $N\times N$ matrix

PX = X * (X' * X)^(-1) * X'

is called the projection matrix.^I.2Subspace projection is an example in which the power of matrix linear algebra notation is evident.