Chapter 10 - The Geometry of Linear and Affine Maps

Section 10.2 - The Decomposition of Linear Maps into Rotations, Reflections, and Rescalings in (R^2)

Commands before Example 10.2.1

(R={{Cos[θ],-Sin[θ]},{Sin[θ],Cos[θ]}})//MatrixForm

(S={{a,0},{0,b}})//MatrixForm

(ReflectX={{1,0},{0,-1}})//MatrixForm

R.S//MatrixForm

S.R//MatrixForm

R.ReflectX//MatrixForm

ReflectX.R//MatrixForm

S.ReflectX//MatrixForm

ReflectX.S//MatrixForm

(TrigFactor[R.(R/.{θ->ϕ})])//MatrixForm

(TrigFactor[(R/.{θ->ϕ}).R])//MatrixForm

(S.(S/.{a-> c,b-> d}))//MatrixForm

((S/.{a->c,b->d}).S)//MatrixForm

Example 10.2.1

ColA1={{-8},{3}};ColA2={{5},{-1}};

R=Norm[ColA1]

Q=Norm[ColA2]

θ=N[π+ArcTan[-3/8]]

{R Cos[θ],R Sin[θ]}

ϕ=N[ArcTan[-1/5]]

{Q Cos[ϕ],Q Sin[ϕ]}

B=1/2 (θ+ϕ)

Cv=1/2 (θ-ϕ)

{{Cos[B],-Sin[B]},{Sin[B],Cos[B]}}.{{Sqrt[2] Cos[Cv],0},{0,Sqrt[2] Sin[Cv]}}.{{Cos[π/4],-Sin[π/4]},{Sin[π/4],Cos[π/4]}}.{{R,0},{0,-Q}}//MatrixForm

{{Cos[B],-Sin[B]},{Sin[B],Cos[B]}}.{{Sqrt[2] Cos[Cv],0},{0,Sqrt[2] Sin[Cv]}}.{{R,0},{0,-Q}}.{{Cos[π/4],-Sin[π/4]},{Sin[π/4],Cos[π/4]}}//MatrixForm