# Section27Orthogonal Projectors (NM, 6 Nov)¶ permalink

This went much better! We covered the following topics.

1. The square matrix $P$ is a projector if $P^2=P\text{.}$ And $v \in \Range(P) \iff Pv=v\text{.}$
2. $I-P$is also a projector.
3. We (finally) noted that $\Null(P)=\Range(I-P)\text{.}$
4. $\Range(P) \cap \Null(P) = \{0\}.$
5. So $P$ separates $\mathbb{C}^m$ into two subspaces, $S_1=\Range(P)$ and $S_2=\Null(p).\text{.}$
6. If $S_1 \perp S_2$ we call $P$ and orthogonal projector. We proved that a projector $P$ is orthogonal if, and only if, $P=P^\star\text{.}$
7. If $\hat{Q}$ is a matrix with orthogonal, orthonormal columns (though not necessarily square, so not unitary), then $P=\hat{Q}\hat{Q}^\star$ is an orthogonal projector.
8. For any vector $q\text{,}$ with $\|q\|_2=1\text{,}$ we can define the projectors \begin{equation*} P_{q}=qq^\star \qquad \text{ and } \qquad P_{\perp q} = I - q q^\star\text{.} \end{equation*} More generally, for any vector \begin{equation*} P_a=\frac{aa^\star}{a^\star a}\text{.} \end{equation*}

# Subsection27.1Exercises

##### Exercise27.1

Show that

\begin{equation*} P = \begin{pmatrix} 0 \amp 0 \\ \alpha \amp 1 \end{pmatrix} \end{equation*}

is a projector. For what $\alpha$ is it an orthogonal projector?

The following three questions are from Lecture 6 of Trefethen and Bau.

##### Exercise27.2

Prove that, if $P$ is an orthogonal projector, then $I-2P$ is a unitary matrix.

##### Exercise27.3

Write down the square matrix $F$ such that

\begin{equation*} F \begin{pmatrix} x_1\\ x_2 \\ \vdots \\ x_m \end{pmatrix} = \begin{pmatrix} x_m\\ x_{n-1} \\ \vdots \\ x_1 \end{pmatrix}. \end{equation*}

Let $E=(I+F)/2\text{.}$ Is $E$ a projector? Is it an orthogonal projector?

##### Exercise27.4

Let $P\in \Cmm$be a non-zero projector. Prove that $\|P\|_2 \geq 1\text{.}$ Prove that $\|P\|_2=1$ if, and only if, $P$ is an orthogonal projector.