Let $A$ be a diagonalizable matrix, and $P$ be permutation matrix of same size. Does $A$ and $PAP$ have the same eigenvalues (or characteristic polynomial)?
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1$\begingroup$ Do you mean $PAP$ or $P^{-1}AP$? $\endgroup$– Binxu Wang 王彬旭May 19, 2022 at 2:13
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1$\begingroup$ Isn't that the same? I mean P is its own inverse, so $PAP$ is the same as $P^{-1} AP$ $\endgroup$– TensorMay 24, 2022 at 4:20
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2$\begingroup$ No thats not the same $P=P^{-1}$ is not always true for permutation matices @HafizAjiAziz $\endgroup$– Binxu Wang 王彬旭May 24, 2022 at 4:58
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1 Answer
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Since $P^\top = P^{-1}$ (it is not the case that $P = P^{-1}$), it follows that the matrices $A$ and $P^\top A P$ have the same characteristic polynomial and hence eigenvalues (the latter is called a permutation similarity).
answered May 24, 2022 at 4:25