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Section27Orthogonal Projectors (NM, 6 Nov)

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*}



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.