HouseholderMatrix - Maple Help

Student[LinearAlgebra]

 HouseholderMatrix
 construct a Householder Matrix

 Calling Sequence HouseholderMatrix(V, options)

Parameters

 V - Vector; used for constructing the Householder Matrix options - (optional) parameters; for a complete list, see LinearAlgebra[HouseholderMatrix]

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right):$
 > $V≔⟨3,4,5⟩$
 ${V}{≔}\left[\begin{array}{c}{3}\\ {4}\\ {5}\end{array}\right]$ (1)
 > $\mathrm{Norm}\left(V\right)$
 ${5}{}\sqrt{{2}}$ (2)
 > $W≔V+\mathrm{Norm}\left(V\right)·⟨1,0,0⟩$
 ${W}{≔}\left[\begin{array}{c}{3}{+}{5}{}\sqrt{{2}}\\ {4}\\ {5}\end{array}\right]$ (3)
 > $H≔\mathrm{HouseholderMatrix}\left(W\right)$
 ${H}{≔}\left[\begin{array}{ccc}{1}{-}\frac{{2}{}{\left({3}{+}{5}{}\sqrt{{2}}\right)}^{{2}}}{{41}{+}{\left({3}{+}{5}{}\sqrt{{2}}\right)}^{{2}}}& {-}\frac{{8}{}\left({3}{+}{5}{}\sqrt{{2}}\right)}{{41}{+}{\left({3}{+}{5}{}\sqrt{{2}}\right)}^{{2}}}& {-}\frac{{10}{}\left({3}{+}{5}{}\sqrt{{2}}\right)}{{41}{+}{\left({3}{+}{5}{}\sqrt{{2}}\right)}^{{2}}}\\ {-}\frac{{8}{}\left({3}{+}{5}{}\sqrt{{2}}\right)}{{41}{+}{\left({3}{+}{5}{}\sqrt{{2}}\right)}^{{2}}}& {1}{-}\frac{{32}}{{41}{+}{\left({3}{+}{5}{}\sqrt{{2}}\right)}^{{2}}}& {-}\frac{{40}}{{41}{+}{\left({3}{+}{5}{}\sqrt{{2}}\right)}^{{2}}}\\ {-}\frac{{10}{}\left({3}{+}{5}{}\sqrt{{2}}\right)}{{41}{+}{\left({3}{+}{5}{}\sqrt{{2}}\right)}^{{2}}}& {-}\frac{{40}}{{41}{+}{\left({3}{+}{5}{}\sqrt{{2}}\right)}^{{2}}}& {1}{-}\frac{{50}}{{41}{+}{\left({3}{+}{5}{}\sqrt{{2}}\right)}^{{2}}}\end{array}\right]$ (4)
 > $\mathrm{evala}\left(H·V\right)$
 $\left[\begin{array}{c}{-}{5}{}\sqrt{{2}}\\ {0}\\ {0}\end{array}\right]$ (5)

Compatibility

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