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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Spring 2022 • Question 3

Question

Math 6610 • Analysis of Numerical Methods I Spring 2022 • Question 3

(Properties of Projectors). Prove algebraically:

  1. Show that if \(P \in \mathbb {C}^{m \times m}\) is a nonzero projector, then \(\|P\|_2 \ge 1\).
  2. Show that if \(P\) is an orthogonal projector, then \(I - 2P\) is unitary.

Raw LaTeX

(Properties of Projectors).
Prove algebraically:
\begin{enumerate}

\item
Show that if \(P \in \mathbb{C}^{m \times m}\) is a nonzero projector, then \(\|P\|_2 \ge 1\).


\item
Show that if \(P\) is an orthogonal projector, then \(I - 2P\) is unitary.

\end{enumerate}

Topic Tags

Linear Programming and Optimization Orthogonal and Unitary Matrices Projectors and Orthogonal Projections