Let A be an n×n matrix whose characteristic polynomial splits. Prove that A and At have the same Jordan canonical form, and conclude that A and At are similar. Hint: For any eigenvalue λ of A and At and any positive integer r, show that rank((A −λI)r) = rank((At−λI)r).

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Asked Dec 31, 2019
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Let A be an n×n matrix whose characteristic polynomial splits. Prove that A and At have the same Jordan canonical form, and conclude that A and At are similar. Hint: For any eigenvalue λ of A and At and any positive integer r, show that rank((A −λI)r) = rank((At−λI)r).

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