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?
This went much better! We covered the following topics.
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.
Prove that, if \(P\) is an orthogonal projector, then \(I-2P\) is a unitary matrix.
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?
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.