Let $A=\begin{bmatrix} cI & P & Q \\ P^\top & cI & R\\ Q^\top & R^\top & cI\\ \end{bmatrix}$ be a block matrix where $P$, $Q$ and $R$ are $n\times n$ matrices with real entries and $c$ is a real number. Let $P_1$, $P_2$ and $P_3$ be $n\times n$ permutation matrices. Let $B= \begin{bmatrix} cI & P_1P & P_2Q \\ (P_1P)^\top & cI & P_3R\\ (P_2Q)^\top & (P_3R)^\top & cI\\ \end{bmatrix}\\$. Is there any relation between the eigenvalues of $A$ and $B$? I have tried using computer programming for obtaining eigenvalues of all possible combinations of $B$ but the output obtained runs to pages and looking for patterns in this is getting extremely tedious. Also, Is there any theory that one could apply for comparing the eigenvalues of A and B?
2026-03-26 12:51:24.1774529484
Relation Between Eigenvalues of Certain Block Matrices
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