# Finding eigenvectors of a rank one projection matrix

From the unit vector $$u=\left(\frac{1}{6},\frac{1}{6},\frac{3}6,\frac{5}6\right)$$ construct the rank one projection matrix $$P=uu^t.$$

• a) Show that if $P=uu^t$ , then $u$ is an eigenvector with $\lambda=1$.
• b) If $v$ is perpendicular to $u$, show that $Pv=0$, then $\lambda=0$
• c) Find three independent eigenvectors of $P$ all with eigenvalue of $\lambda=0$.
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Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. If this is homework, please add the homework tag; people will still help, so don't worry. Also, many find the use of imperative ("Prove", "Solve", etc.) to be rude when asking for help; please consider rewriting your post. – Julian Kuelshammer Nov 20 '12 at 20:07
Before all the downvotes, let's let John try and clarify, edit, respond to the message Julian left. After all, the question was posted only minutes ago! And John is new. Cut some slack! – amWhy Nov 20 '12 at 20:13
@Julian. It's not the imperative as a grammatical device that's rude -- the standard style of written mathematics uses imperatives all over the place. What's rude is writing a question that looks like it consists only of a verbatim quote from an exercise sheet with no thought of the asker's own to go with it. – Henning Makholm Nov 20 '12 at 20:31
@Henning: You should improve the comment template then. :) (Although I think imperative style in written mathematics is only used when the author is assigning the reader an exercise. Certainly it is out of place when asking for help, here or elsewhere.) – Rahul Nov 20 '12 at 23:11
@RahulNarain: Do you think every instance of "Let $v_k=\cdots$", "Consider the function $f(x)=\cdots$", "Proceed by induction on $|Q|$", "Assume first that $\phi$ does not hold", or "Now apply lemma 5.2.6 to $w$ and $\alpha'$" constitute assigning the reader an exercise? Certainly the first two of these look completely appropriate when asking for help, here and elsewhere. – Henning Makholm Nov 21 '12 at 13:52

## 1 Answer

1. $Pu$ is an eigenvector means that $Pu = \lambda$. We have $Pu = uu^t u$, which equals what?
2. $v$ perpendicular to $u$ means $u^t v = 0$. Now compute $Pv$.
3. By using the above problem, finding three vectors $v_1, v_2, v_3$ such that $Pv_i = 0$ for each $i$ is the same as finding three vectors each perpendicular to $u$. See if you can set up an equation to solve for all vectors perpendicular to $u$.
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