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Question Number 346 by 123456 last updated on 25/Jan/15
Γ(θ)= [((cos θ),(sin θ)),((−sin θ),(cos θ)) ] Λ(θ,t)= [((cos θ),(sinh t sin θ)),((sin θ),(cosh t cos θ)) ] ζ(θ,t)=Γ(θ)×Λ(θ,t)+Λ(θ,t)×Γ(θ) ζ(θ,0)=? det ζ(θ,0)=?
$$\Gamma\left(\theta\right)=\begin{bmatrix}{\mathrm{cos}\:\theta}&{\mathrm{sin}\:\theta}\\{−\mathrm{sin}\:\theta}&{\mathrm{cos}\:\theta}\end{bmatrix} \\ $$$$\Lambda\left(\theta,{t}\right)=\begin{bmatrix}{\mathrm{cos}\:\theta}&{\mathrm{sinh}\:{t}\:\mathrm{sin}\:\theta}\\{\mathrm{sin}\:\theta}&{\mathrm{cosh}\:{t}\:\mathrm{cos}\:\theta}\end{bmatrix} \\ $$$$\zeta\left(\theta,{t}\right)=\Gamma\left(\theta\right)×\Lambda\left(\theta,{t}\right)+\Lambda\left(\theta,{t}\right)×\Gamma\left(\theta\right) \\ $$$$\zeta\left(\theta,\mathrm{0}\right)=? \\ $$$$\mathrm{det}\:\zeta\left(\theta,\mathrm{0}\right)=? \\ $$
Answered by prakash jain last updated on 23/Dec/14
Λ(θ,0)= [((cos θ),0),((sin θ),(cos θ)) ] Γ(θ)×Λ(θ,0)= [((cos^2 θ+sin^2 θ),(0+cos θsin θ)),((−sin θcos θ+sin θcos θ),(cos^2 θ)) ] = [(1,(cos θsin θ)),(0,(cos^2 θ)) ] Λ(θ,0)×Γ(θ)= [((cos^2 θ+0),(cos θsin θ+0)),((cos θsin θ−sin θcos θ),1) ] = [((cos^2 θ),(cos θsin θ)),(0,1) ] Γ(θ)×Λ(θ,0)+Λ(θ,0)×Γ(θ)= [((1+cos^2 θ),(2cos θsin θ)),(0,(1+cos^2 θ)) ] ζ(θ,0)= [((1+cos^2 θ),(2cos θsin θ)),(0,(1+cos^2 θ)) ] det ζ(θ,0)=(1+cos^2 θ)^2
$$\Lambda\left(\theta,\mathrm{0}\right)=\begin{bmatrix}{\mathrm{cos}\:\theta}&{\mathrm{0}}\\{\mathrm{sin}\:\theta}&{\mathrm{cos}\:\theta}\end{bmatrix} \\ $$$$\Gamma\left(\theta\right)×\Lambda\left(\theta,\mathrm{0}\right)=\begin{bmatrix}{\mathrm{cos}^{\mathrm{2}} \theta+\mathrm{sin}^{\mathrm{2}} \theta}&{\mathrm{0}+\mathrm{cos}\:\theta\mathrm{sin}\:\theta}\\{−\mathrm{sin}\:\theta\mathrm{cos}\:\theta+\mathrm{sin}\:\theta\mathrm{cos}\:\theta}&{\mathrm{cos}^{\mathrm{2}} \theta}\end{bmatrix} \\ $$$$=\begin{bmatrix}{\mathrm{1}}&{\mathrm{cos}\:\theta\mathrm{sin}\:\theta}\\{\mathrm{0}}&{\mathrm{cos}^{\mathrm{2}} \theta}\end{bmatrix} \\ $$$$\Lambda\left(\theta,\mathrm{0}\right)×\Gamma\left(\theta\right)=\begin{bmatrix}{\mathrm{cos}^{\mathrm{2}} \theta+\mathrm{0}}&{\mathrm{cos}\:\theta\mathrm{sin}\:\theta+\mathrm{0}}\\{\mathrm{cos}\:\theta\mathrm{sin}\:\theta−\mathrm{sin}\:\theta\mathrm{cos}\:\theta}&{\mathrm{1}}\end{bmatrix} \\ $$$$=\begin{bmatrix}{\mathrm{cos}^{\mathrm{2}} \theta}&{\mathrm{cos}\:\theta\mathrm{sin}\:\theta}\\{\mathrm{0}}&{\mathrm{1}}\end{bmatrix} \\ $$$$\Gamma\left(\theta\right)×\Lambda\left(\theta,\mathrm{0}\right)+\Lambda\left(\theta,\mathrm{0}\right)×\Gamma\left(\theta\right)=\begin{bmatrix}{\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \theta}&{\mathrm{2cos}\:\theta\mathrm{sin}\:\theta}\\{\mathrm{0}}&{\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \theta}\end{bmatrix} \\ $$$$\zeta\left(\theta,\mathrm{0}\right)=\begin{bmatrix}{\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \theta}&{\mathrm{2cos}\:\theta\mathrm{sin}\:\theta}\\{\mathrm{0}}&{\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \theta}\end{bmatrix} \\ $$$$\mathrm{det}\:\zeta\left(\theta,\mathrm{0}\right)=\left(\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \theta\right)^{\mathrm{2}} \\ $$

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