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Given $G=\left\{A\in M_2(\mathbb{R})\mid A^\top XA = X\right\}$.

Need to find the basis. Error in question

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Treat $Y^TX+XY=0$ as a system of homogeneous equations in the unknown four entries $y_{ij}$ of the matrix $Y$. Use linear algebra techniques to find a basis for the solution space. I got a different basis (but a 1-dimensional space anyway). –  Jyrki Lahtonen Dec 31 '11 at 9:51

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Let $$ Y=\pmatrix{a&b\cr c&d\cr}. $$ Recall that (from the earlier questions?) $$ X=\pmatrix{3&1\cr 1&1\cr}. $$ The matrix equation defining the Lie algebra looks like $$ Y^TX+XY=\pmatrix{6a+2c&a+3b+c+d\cr a+3b+c+d&2b+2d\cr}=0. $$ This is a system of 4 linear homogeneous equations (two of them are the same, though). The bottom right corner tells you that $b=-d$. The top left tells you that $c=-3a$. Making these substitution in the remaining equation gives $0=a-3d-3a+d=-2a-2d,$ so $a=-d$, and hence $c=-3a=3d$. Putting it all together you get $$Y=d\left(\begin{array}{rr}-1&-1\\ 3&1\end{array}\right).$$ The Lie algebra is thus 1-dimensional. It is spanned by the above matrix.

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