Householder transformation calculation

Question

Let $H=I-2ww^t$ be the Householder transformation such that

$H\begin{bmatrix}-24\\0\\7\end{bmatrix}=\begin{bmatrix}25\\0\\0\end{bmatrix}$

what is w?

Now I know H is involutory and orthogonal so $H^2=H^TH=I$ and H is symmetric $H=H^T$. So I can have HA=B and A=HB. That will give me a lot of equations to solve. Is there another (shorter) way of doing this?

Answer 1

Well, from this equation you see that

$$ -49 = 2w_1(-24w_1 + 7w_3), \quad 0 = 2w_2(-24w_1 + 7w_3), \quad\text{and}\quad 7 = 2w_3(-24w_1 + 7w_3) $$

Multiplying the first equation by $w_3$ and the third equation by $w_1$ it follows that $w_1 = -7w_3$. Thus you get

$$ w_3 = \pm \frac{1}{\sqrt{50}} \implies w_1 = \mp\frac{7}{\sqrt{50}} \quad\text{and}\quad w_2 = 0 $$

Answer 2

Householder transformation is a reflection. Reflections are by definition linear transformations that send one specific vector (and its scalar multiples) to its negative, and all vectors that are orthogonal to it are undisturbed. (So one eigenvalue will be $-1$ and $n-1$ eigenvalues will be $+1$, when operating on $\mathbf{R}^n$.)

The formula you have written is the matrix form of a reflection that sends a unit length vector $w$ to its negative.