Student[LinearAlgebra] - Maple Programming Help

Home : Support : Online Help : Education : Student Package : Linear Algebra : Computation : Constructors : Student/LinearAlgebra/ReflectionMatrix

Student[LinearAlgebra]

 ReflectionMatrix
 construct a reflection Matrix

 Calling Sequence ReflectionMatrix(v)

Parameters

 v - Vector; normal to the reflection subspace

Description

 • The ReflectionMatrix(v) command returns the $mxm$ reflection Matrix, R, determined by the Vector v, where $m=\mathrm{Dimension}\left(v\right)$. Thus $R\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}v=-v$ and $R\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}w=w$ for all w such that $v\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}w=0$.
 • In two-dimensional space, the reflection subspace is a line through the origin.  If the equation of this line is $y=mx$, or $mx-y=0$, then $v=⟨m,-1⟩$.
 In three-dimensional space, the reflection subspace is a plane through the origin.

Examples

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

Reflect through the line y=-x:

 > $\mathrm{ReflectionMatrix}\left(⟨1,1⟩\right)$
 $\left[\begin{array}{rr}{0}& {-}{1}\\ {-}{1}& {0}\end{array}\right]$ (1)
 > $\mathrm{ReflectionMatrix}\left(⟨a,b,c⟩\right)$
 $\left[\begin{array}{ccc}\frac{{-}{{a}}^{{2}}{+}{{b}}^{{2}}{+}{{c}}^{{2}}}{{{a}}^{{2}}{+}{{b}}^{{2}}{+}{{c}}^{{2}}}& {-}\frac{{2}{}{a}{}{b}}{{{a}}^{{2}}{+}{{b}}^{{2}}{+}{{c}}^{{2}}}& {-}\frac{{2}{}{a}{}{c}}{{{a}}^{{2}}{+}{{b}}^{{2}}{+}{{c}}^{{2}}}\\ {-}\frac{{2}{}{a}{}{b}}{{{a}}^{{2}}{+}{{b}}^{{2}}{+}{{c}}^{{2}}}& \frac{{{a}}^{{2}}{-}{{b}}^{{2}}{+}{{c}}^{{2}}}{{{a}}^{{2}}{+}{{b}}^{{2}}{+}{{c}}^{{2}}}& {-}\frac{{2}{}{b}{}{c}}{{{a}}^{{2}}{+}{{b}}^{{2}}{+}{{c}}^{{2}}}\\ {-}\frac{{2}{}{a}{}{c}}{{{a}}^{{2}}{+}{{b}}^{{2}}{+}{{c}}^{{2}}}& {-}\frac{{2}{}{b}{}{c}}{{{a}}^{{2}}{+}{{b}}^{{2}}{+}{{c}}^{{2}}}& \frac{{{a}}^{{2}}{+}{{b}}^{{2}}{-}{{c}}^{{2}}}{{{a}}^{{2}}{+}{{b}}^{{2}}{+}{{c}}^{{2}}}\end{array}\right]$ (2)
 > $\mathrm{ReflectionMatrix}\left(⟨1,2,3,4⟩\right)$
 $\left[\begin{array}{cccc}\frac{{14}}{{15}}& {-}\frac{{2}}{{15}}& {-}\frac{{1}}{{5}}& {-}\frac{{4}}{{15}}\\ {-}\frac{{2}}{{15}}& \frac{{11}}{{15}}& {-}\frac{{2}}{{5}}& {-}\frac{{8}}{{15}}\\ {-}\frac{{1}}{{5}}& {-}\frac{{2}}{{5}}& \frac{{2}}{{5}}& {-}\frac{{4}}{{5}}\\ {-}\frac{{4}}{{15}}& {-}\frac{{8}}{{15}}& {-}\frac{{4}}{{5}}& {-}\frac{{1}}{{15}}\end{array}\right]$ (3)