Householder reflection sending a vector to a coordinate axis ($Hx=\alpha e_1,|\alpha |=\Vert x{\Vert }_2$)

For a nonzero real vector $x\in {\mathbb{R}}^n$, choose $\alpha$ with $|\alpha |=\Vert x{\Vert }_2$ and put $v=x-\alpha e_1$. When $v\ne 0$, the Householder reflectionsatisfies $Hx=\alpha e_1$: every component except the first becomes zero. Here $e_1$ is the first Cartesian basis vector. The calculation is short:If $v=0$, the vector is already at the selected target, so the identity matrix suffices.

In floating-point arithmetic, a customary choice isThen $|v_1|=|x_1|+\Vert x{\Vert }_2$, avoiding subtraction of nearly equal positive quantities in the first component. Either sign can be valid in exact arithmetic; the numerical choice protects the construction from catastrophic cancellation. ^[ref]

For $x=(3,4{)}^T$, this choice gives $\alpha =-5$ and $v=(8,4{)}^T$. Dividing $v$ by its Euclidean norm gives $u=(2,1{)}^T/\sqrt{5}$, exactly the normal in the geometric example.