Projector

Let $$\mathcal X$$ and $$\mathcal Y$$ be complementary subspaces of a vector space $$V$$ so that each $$\mathbf v \in \mathcal V$$ can be uniquely resolved as $$\mathbf{v = x+y}$$, where $$\mathbf x \in \mathcal X$$ and $$\mathcal Y$$. The unique linear operator $$\mathbf P$$ is called the projector onto $$\mathcal X$$ along $$\mathcal Y$$, and $$\mathbf P$$ has the following properties (p386 in ):


 * $$\mathbf P^2 = \mathbf P$$
 * $$\mathbf{I-P}$$ is the complementary projector onto $$\mathcal Y$$ along $$\mathcal X$$.
 * $$R(\mathbf P) = \{\mathbf x : \mathbf{Px = x}\}$$   <-- this is a property used frequently in proofs
 * $$R(\mathbf P) = N (\mathbf{I-P}) = \mathcal X$$ and $$R(\mathbf{I-P})=N(\mathbf P) = \mathcal Y$$
 * If $$\mathcal V = \mathbn R^n or \mathbb C^n$$, then $$\mathbf P$$is given by

$$\mathbf P = [\mathbf X | \mathbf 0] [\mathbf X | \mathbf Y]^{-1} = [\mathbf X | \mathbf Y] \begin{pmatrix} \mathbf I & \matbhf 0 \\ \mathbf 0 & \mathbf 0 \end{pmatrix} [\mathbf X | \mathbf Y]^{-1}$$