-Mohr
-使Mohrtakker
A:=(a00a01a01a11)A:=\begin{pmatrix}a_{00}&a_{01}\\a_{01}&a_{11}\end{pmatrix}Mohr

AA-3
A=(aa01a01a)+a+(1001)A=\begin{pmatrix}a_-&a_{01}\\a_{01}&-a_-\end{pmatrix}+a_+\begin{pmatrix}1&0\\0&1\end{pmatrix}
a:=12(a00a11)a_-:=\frac12(a_{00}-a_{11})
a+:=12(a00+a11)a_+:=\frac12(a_{00}+a_{11})

R=(abba)R=\begin{pmatrix}a&-b\\b&a\end{pmatrix}J=(0110)J=\begin{pmatrix}0&1\\1&0\end{pmatrix}
RJ=JRRJ=JR^\top
RJ=(abba)(0110)=(baab)RJ=\begin{pmatrix}a&-b\\b&a\end{pmatrix}\begin{pmatrix}0&1\\1&0\end{pmatrix}=\begin{pmatrix}-b&a\\a&b\end{pmatrix}
RJ=(RJ)=JR=JR\underline{RJ=(RJ)^\top=J^\top R^\top=JR^\top\quad}_\blacksquare

θ\thetaAA
A:=(a00a01a01a11):=R(θ)AR(θ)A':=\begin{pmatrix}a'_{00}&a'_{01}\\a'_{01}&a'_{11}\end{pmatrix}:={R(\theta)}^\top AR(\theta)
R(θ):=(cosθsinθsinθcosθ)R(\theta):=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}
A=R(θ)AR(θ)A'={R(\theta)}^\top AR(\theta)
=R(θ)(aa01a01a)R(θ)+a+R(θ)IR(θ)={R(\theta)}^\top\begin{pmatrix}a_-&a_{01}\\a_{01}&-a_-\end{pmatrix}{R(\theta)}+a_+{R(\theta)}^\top IR(\theta)
\because①
=R(θ)(a01aaa01)(0110)R(θ)+a+I=R(-\theta)\begin{pmatrix}a_{01}&a_-\\-a_-&a_{01}\end{pmatrix}\begin{pmatrix}0&1\\1&0\end{pmatrix}{R(\theta)}+a_+ I
=a012+a2R(θ)R(ϕ12π)R(θ)(0110)+a+I=\sqrt{{a_{01}}^2+{a_-}^2}R(-\theta)R\left(\phi-\frac12\pi\right)R(-\theta)\begin{pmatrix}0&1\\1&0\end{pmatrix}+a_+ I
\because②
ϕ:=tan1a01a\phi:=\tan^{-1}\frac{a_{01}}{a_-}
a012+a2cosϕ=a,a012+a2sinϕ=a01\sqrt{{a_{01}}^2+{a_-}^2}\cos\phi=a_-,\sqrt{{a_{01}}^2+{a_-}^2}\sin\phi=a_{01}
cos(ϕ12π)=(ieiϕ)=sinϕ\cos(\phi-\frac12\pi)=\Re(-ie^{i\phi})=\sin\phi
sin(ϕ12π)=(ieiϕ)=cosϕ\sin(\phi-\frac12\pi)=\Im(-ie^{i\phi})=-\cos\phi
=a012+a2R(ϕ2θ12π)(0110)+a+I=\sqrt{{a_{01}}^2+{a_-}^2}R\left(\phi-2\theta-\frac12\pi\right)\begin{pmatrix}0&1\\1&0\end{pmatrix}+a_+ I
\because ②
=a012+a2(sin(ϕ2θ)cos(ϕ2θ)cos(ϕ2θ)sin(ϕ2θ))(0110)+a+I=\sqrt{{a_{01}}^2+{a_-}^2}\begin{pmatrix}\sin(\phi-2\theta)&\cos(\phi-2\theta)\\-\cos(\phi-2\theta)&\sin(\phi-2\theta)\end{pmatrix}\begin{pmatrix}0&1\\1&0\end{pmatrix}+a_+ I
    {a00=a012+a2cos(ϕ2θ)+a+a01=a012+a2sin(ϕ2θ)\underline{\implies\begin{dcases}a'_{00}&=\sqrt{{a_{01}}^2+{a_-}^2}\cos(\phi-2\theta)+a_+\\a'_{01}&=\sqrt{{a_{01}}^2+{a_-}^2}\sin(\phi-2\theta)\end{dcases}\quad}_\blacksquare
(a00,a01)(a'_{00},a'_{01})Mohr

R(θ)AR(θ){R(\theta)}^\top AR(\theta)

3
a+:=13(a00+a11+a22)a_+:=\frac13(a_{00}+a_{11}+a_{22})
a01:=13(a00a11)a_{0-1}:=\frac13(a_{00}-a_{11})
a12:=13(a11a22)a_{1-2}:=\frac13(a_{11}-a_{22})
a21:=13(a22a00)a_{2-1}:=\frac13(a_{22}-a_{00})

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