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When I read an article by Garbe, where have an eigenmatrix $$B M_i + M_i B^T=\lambda_iM_i$$ and have the following eigenmatrices and eigenvalues:

$$ M_0=\begin{bmatrix} 0 & 1 \\ -1 & 0\end{bmatrix},M_1=\begin{bmatrix} \frac{1}{\sqrt{Xg^2-1}} & 0 \\ 0 & -\sqrt{Xg^2-1}\end{bmatrix}, M_{\pm}=\begin{bmatrix} \frac{1}{\sqrt{Xg^2-1}} & \pm1 \\ \pm1 & \sqrt{Xg^2-1}\end{bmatrix}\\ \lambda_0=\lambda_1=0, \lambda_{\pm}=\pm2\omega_0\sqrt{Xg^2-1}$$

I tried to solve the matrix equation, hoping to get the same result as the article, but I didn't have any beneficial results.

Below is my code for the question

B={{0,w0},{w0(X g^2-1),0}} M=Array[m,{2,2}] \[Lambda]={{\[Lambda]1,\[Lambda]2},{\[Lambda]3,\[Lambda]4}} sol=Solve[B.M+M.B\[Transpose]==\[Lambda] M,Flatten[M]] 

It's not producing the results I want.

Any help is much appreciated.

When I read an article by Garbe, where have an eigenmatrix $$B M_i + M_i B^T=\lambda_iM_i$$ and have the following eigenmatrices and eigenvalues:

$$ M_0=\begin{bmatrix} 0 & 1 \\ -1 & 0\end{bmatrix},M_1=\begin{bmatrix} \frac{1}{\sqrt{Xg^2-1}} & 0 \\ 0 & -\sqrt{Xg^2-1}\end{bmatrix}, M_{\pm}=\begin{bmatrix} \frac{1}{\sqrt{Xg^2-1}} & \pm1 \\ \pm1 & \sqrt{Xg^2-1}\end{bmatrix}\\ \lambda_0=\lambda_1=0, \lambda_{\pm}=\pm2\omega_0\sqrt{Xg^2-1}$$

I tried to solve the matrix equation, hoping to get the same result as the article, but I didn't have any beneficial results.

Below is my code for the question

B = {{0, w0}, {w0(X g^2-1), 0}} M = Array[m, {2, 2}] λ = {{λ1, λ2}, {λ3, λ4}} sol = Solve[B.M + M.B\[Transpose] == λ M, Flatten[M]] 

It's not producing the results I want.

Any help is much appreciated.

When i read an article, where have an eigenmatrix $$B M_i + M_i B^T=\lambda_iM_i$$ and have the following eigenmatrices and eigenvalues:

$$ M_0=\begin{bmatrix} 0 & 1 \\ -1 & 0\end{bmatrix},M_1=\begin{bmatrix} \frac{1}{\sqrt{Xg^2-1}} & 0 \\ 0 & -\sqrt{Xg^2-1}\end{bmatrix}, M_{\pm}=\begin{bmatrix} \frac{1}{\sqrt{Xg^2-1}} & \pm1 \\ \pm1 & \sqrt{Xg^2-1}\end{bmatrix}\\ \lambda_0=\lambda_1=0, \lambda_{\pm}=\pm2\omega_0\sqrt{Xg^2-1}$$

I tried to solve the matrix equation, hoping to get the same result as the article, but I didn't have any beneficial results.

Below is my code for the question

B={{0,w0},{w0(X g^2-1),0}} M=Array[m,{2,2}] \[Lambda]={{\[Lambda]1,\[Lambda]2},{\[Lambda]3,\[Lambda]4}} sol=Solve[B.M+M.B\[Transpose]==\[Lambda] M,Flatten[M]] 

It's not producing the results I want.

Any help is much appreciated.

When I read an article by Garbe, where have an eigenmatrix $$B M_i + M_i B^T=\lambda_iM_i$$ and have the following eigenmatrices and eigenvalues:

$$ M_0=\begin{bmatrix} 0 & 1 \\ -1 & 0\end{bmatrix},M_1=\begin{bmatrix} \frac{1}{\sqrt{Xg^2-1}} & 0 \\ 0 & -\sqrt{Xg^2-1}\end{bmatrix}, M_{\pm}=\begin{bmatrix} \frac{1}{\sqrt{Xg^2-1}} & \pm1 \\ \pm1 & \sqrt{Xg^2-1}\end{bmatrix}\\ \lambda_0=\lambda_1=0, \lambda_{\pm}=\pm2\omega_0\sqrt{Xg^2-1}$$

I tried to solve the matrix equation, hoping to get the same result as the article, but I didn't have any beneficial results.

Below is my code for the question

B={{0,w0},{w0(X g^2-1),0}} M=Array[m,{2,2}] \[Lambda]={{\[Lambda]1,\[Lambda]2},{\[Lambda]3,\[Lambda]4}} sol=Solve[B.M+M.B\[Transpose]==\[Lambda] M,Flatten[M]] 

It's not producing the results I want.

Any help is much appreciated.