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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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  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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