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I want to find the eigenvalues of the following $6\times6$ block matrix:

\begin{pmatrix} A & B \\ B & C \end{pmatrix} having zero diagonal terms, where each block is a $3\times3$ matrix of given eigenvalues, Is their any property I can use?

mohammad farhat
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    There is no known correspondence unless $B=0$ (in this case, take the union of eigenvalues of $A$ and $C$). – Jean Marie Jan 02 '21 at 22:57

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It is not determined by the eigenvalues of the blocks. Consider $$\pmatrix{0 & 1 & 1 & -2 \sin(t) \cos(t) & 2 \cos^2(t)-1 & \cos(t)-\sin(t)\cr 1 & 0 & 1 & 2 \cos^2(t)-1 & 2 \sin(t) \cos(t) & \sin(t)+\cos(t)\cr 1 & 1 & 0 & \cos(t)-\sin(t) & \sin(t)+\cos(t) & 0\cr -2 \sin(t) \cos(t) & 2 \cos^2(t)-1 & \cos(t) - \sin(t) & 0 & 1 & 1\cr 2 \cos^2(t)-1 & 2 \sin(t) \cos(t) & \sin(t)+\cos(t) & 1 & 0 & 1\cr \cos(t)-\sin(t) & \sin(t)+\cos(t) & 0 & 1 & 1 & 0} $$ where $A,B,C$ all have the same eigenvalues $2, -1, -1$ but the block matrix has characteristic polynomial $\lambda (\lambda + 2)(\lambda -2 - 2 \cos(t)) (\lambda + 2 \cos(t))(\lambda^2 + 4 \cos^2(t) + 4 \cos(t) - 8)$ that depends on $t$.

Robert Israel
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