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If a $10\times10$ matrix which consists of two $5\times5$ diagonal blocks on the upper right corner and lower left corner (and all other entries are zero) is related by a similarity transformation to another matrix, what can you say about the eigenvalues of the blocks and the eigenvalues of the other matrix?

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Suppose $A = \pmatrix{0 & B\cr C & 0\cr}$ where $B$ and $C$ are your diagonal blocks. Similar matrices have the same eigenvalues, so the question is what are the eigenvalues of $A$. Note that if $$ A \pmatrix{u \cr v\cr} = \pmatrix{B v\cr C u\cr} = \lambda \pmatrix{u\cr v\cr}$$ we have $Bv = \lambda u$ and $C u = \lambda v$.
It's not hard to show that the eigenvalues of $A$ are $\pm \sqrt{B_{ii} C_{ii}}$.

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