Chapter 7.5, Problem 25E

Consider a $\left(4\times 4\right)$ matrix $B$ of the form shown in $\left(9\right)$, where $b_{31}=0$ and $b_{41}=0$. Let $u$ be a vector of the form $u={[0,a,b,c]}^T$, and let $Q=I-bu{u}^T$ be the associated Householder matrix.

a) Show that forming the product $BQ$ does not change the first column $B$. [Hint: Form $BQ$ by using the two-step procedure illustrated in Example $5$.]

b) Let $u$ be a vector of the form $u={[0,0,a,b]}^T$, and let $Q=I-bu{u}^T$ be the associated Householder matrix. Show that forming $QBQ$ does not alter the first column of $B$.