 Solution

Let $P=u{u}^T/\Vert u{\Vert }^2$. This is an orthogonal projection matrix, so $P^T=P$ and $P^2=P$. Therefore the Householder transformationsatisfies $H_u^T{H}_u=(I-2P{)}^2=I$. If $\Vert a\Vert =\Vert b\Vert$ and $a\ne b$, thenand hence $H_{a-b}a=a-(a-b)=b$.

Apply a Householder transformation to the first column of $A$ to map it to a multiple of $e_1$, then apply transformations supported on the trailing coordinates to clear each later column below its diagonal. Their product is orthogonal and produces $R$; reversing the product gives $A=QR$.

For the displayed matrix, one resulting factorization isDirect multiplication gives the stated $A$, and $Q^{T}Q=I$.