Exercise 6.5.1

Answers

$P^{2} = \frac{a a^{T}}{a^{T} a} \frac{a a^{T}}{a^{T} a} = \frac{a a^{T} a a^{T}}{a^{T} a a^{T} a} = \frac{a a^{T} a^{T} a}{a^{T} a a^{T} a} = \frac{a a^{T}}{a^{T} a} = P$
$Px = \frac{a a^{T}}{a^{T} a} x = \frac{a a^{T} x}{a^{T} a} = \frac{a^{T} xa}{a^{T} a}$ , note $a^{T} x$ is a scalar, so $Px$ is in the direction of $a$ .
$a^{T} (x - Px) = a^{T} x - a^{T} Px = a^{T} x - a^{T} \frac{a a^{T}}{a^{T} a} x = a^{T} x - \frac{a^{T} a a^{T} x}{a^{T} a} = a^{T} x - a^{T} x = 0$

niuers

2020-03-20 00:00