Orthogonal Projection

linear-algebra

Definition

Orthogonal Projection

Let $P:\mathcal{H}\to \mathcal{H}$ be a linear operator on an inner product space $\mathcal{H}$. The operator is an orthogonal projection if it is Hermitian and idempotent, that is,

$$P=P^{†}\text{and}{P}^2=P$$

Equivalently, $P$ projects each vector onto a subspace along its orthogonal complement.

Example

The operator $T:{\mathbb{C}}^2\to C^2$, given by the matrix:

$$A=\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)$$

is an orthogonal projection.

In particular, $T$ corresponds to the projection to the first component:

$$T\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)=\left(\begin{array}{c}{x}_1\\ 0\end{array}\right)$$