numerical linear algebra reflectors

Question

This is an intro to numerical linear algebra course. I am not sure how to start with this proof.

Let $u \in \mathbb{R}^n$ and suppose that $\| u \|_2 = 1$. If $P=uu^\top$, prove that

a.)$p^2=p$

b.)$p^\top = p$

What does this question have to do with reflectors? – Ben Grossmann May 19 '17 at 15:14 — Ben Grossmann, May 19 '17 at 15:14

score 1 · Answer 1 · answered May 19 '17 at 14:54

1

Just recall that $$\Vert u\Vert_2^2 =u^\top u$$ and and that $$(uv)^\top=v^\top u^\top$$ and the result follows easily.

answered May 19 '17 at 14:54

user296113

for part a, it says that $|u|_2=1$ not $|u|^2_2=1$. So I am not sure how to proceed. – user123456 May 19 '17 at 15:00
If we have $x\ge 0$ and $x^2=1$ then $x=1$. – user296113 May 19 '17 at 15:02
ah okay I understand now, thanks!. Is there any chance that you can help me start with this last problem that I need. 3.) If $Q$ is an orthogonal matrix, prove that $| Q |_2 = | Q^{-1} |_2 = 1$. – user123456 May 19 '17 at 15:06
I know that $QQ^\top = I$. I am not sure how I can come up with a 1 as a result. – user123456 May 19 '17 at 15:08
@RenzoVallejos if you have a new question, you should really be making a new post – Ben Grossmann May 19 '17 at 15:15

1 Answers1