Householder QR derivation

Question

Im looking at the Householder operation, does anyone know how this guy got from the first equation to the second?

$$\forall x, Px = x - \frac{2v(x^Tv)}{v^Yv} \implies P = I-\frac{2vv^T}{v^Tv}$$

I know this is probably very simple but I am very new to matrices and all their defined properties and operations

Note that for vectors $x^T v = v^T x$. – lisyarus Sep 09 '20 at 12:44 — lisyarus, Sep 09 '20 at 12:44
Ahhh ok thank you didnt knwo that – Hyper Relic Sep 09 '20 at 12:49 — Hyper Relic, Sep 09 '20 at 12:49

score 3 · Answer 1 · answered Sep 09 '20 at 12:46

$$\forall x , Px=x-\frac{2v(x^Tv)}{v^Tv}=Ix -\frac{2v(v^Tx)}{v^Tv} =\left(I-\frac{2vv^T}{v^Tv}\right)x$$

$$\forall x , \left( P- \left(I-\frac{2vv^T}{v^Tv}\right)\right)x=0$$

The nullspace is $\mathbb{R}^n$, we must have $$P- \left(I-\frac{2vv^T}{v^Tv}\right)=0$$

$$P= \left(I-\frac{2vv^T}{v^Tv}\right)$$

1 Answers1