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I was told that there can't exist a matrix $M\in M_3(\mathbb R)$ such that $M^TCM$ and $M^TDM$ are both diagonal,where $$C=\begin{pmatrix}1&0&0\\0&-1&0\\ 0&0&0\end{pmatrix}$$ and $$D=\begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix}.$$ Why is this necessarily true? Can this be shown without actually writing out $$M=\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}$$ and dealing with messy algebra? I am quite sure there is a simpler way...

Added: where the resulting matrices are not the zero matrix

Thanks.

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    Have you tried $M=O$ or $M$ a matrix with only non zero entry in the lower right corner?2011-12-03
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    @Jan: Thanks, I should have been clearer, my bad, but I am looking for the resultant diagonal matrices to be non-trivial. -- so at least one non-zero entry along the diagonal.2011-12-03

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