The HouseholderMatrix(V) function constructs a Householder Matrix from the values in the Vector V.

If $V$ is a Vector of dimension n and $V^{+}$ indicates its transpose, then HouseholderMatrix(V) constructs the Matrix $\mathrm{Id}-sV.V^{+}$ (if V is a column Vector) or $\mathrm{Id}-s{V}^{+}.V$ (if V is a row Vector), where $\mathrm{Id}=\mathrm{IdentityMatrix}\left(n\right)$ and  $s=\frac{2}{V.V}$ by default.  You can provide a different value for $s$.  See LinearAlgebra[HouseholderMatrix] for more information.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right)\:$

$V≔⟨3\,4\,5⟩$

$V≔\left[\begin{array}{c}3\\ 4\\ 5\end{array}\right]$

$\mathrm{Norm}\left(V\right)$

$5\sqrt{2}$

$W≔V+\mathrm{Norm}\left(V\right)·⟨1\,0\,0⟩$

$W≔\left[\begin{array}{c}3+5\sqrt{2}\\ 4\\ 5\end{array}\right]$

$H≔\mathrm{HouseholderMatrix}\left(W\right)$

$H≔\left[\begin{array}{ccc}1-\frac{2{\left(3+5\sqrt{2}\right)}^2}{41+{\left(3+5\sqrt{2}\right)}^2}& -\frac{8\left(3+5\sqrt{2}\right)}{41+{\left(3+5\sqrt{2}\right)}^2}& -\frac{10\left(3+5\sqrt{2}\right)}{41+{\left(3+5\sqrt{2}\right)}^2}\\ -\frac{8\left(3+5\sqrt{2}\right)}{41+{\left(3+5\sqrt{2}\right)}^2}& 1-\frac{32}{41+{\left(3+5\sqrt{2}\right)}^2}& -\frac{40}{41+{\left(3+5\sqrt{2}\right)}^2}\\ -\frac{10\left(3+5\sqrt{2}\right)}{41+{\left(3+5\sqrt{2}\right)}^2}& -\frac{40}{41+{\left(3+5\sqrt{2}\right)}^2}& 1-\frac{50}{41+{\left(3+5\sqrt{2}\right)}^2}\end{array}\right]$

$\mathrm{evala}\left(H·V\right)$

$\left[\begin{array}{c}-5\sqrt{2}\\ 0\\ 0\end{array}\right]$

The Student[LinearAlgebra][HouseholderMatrix] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.