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This is question 1.6 from Humphrey's "Introduction to Lie Algebras and Representation Theory":

Suppose $x \in \mathfrak{gl}(n,\mathrm{F})$ has $n$ distinct eigenvalues $a_1,\ldots,a_n$. The eigenvalues of the adjoint representation $\mathrm{ad}(x)$ are then precisely the $n^2$ scalars $a_i - a_j$ for $1\leq i,j \leq n$, which of course need not to be distinct.

How do I go on proving this? Any help will be much appreciated

Marcel S
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  • Thank you! When I searched for it, I couldn't find the duplicate unfortunately... – Marcel S Sep 04 '17 at 12:00
  • No problem, sometimes questions are hard to find on the site but once you ask it, the question pops out in the "Related" section on the right side (that's where I saw it). – levap Sep 04 '17 at 12:07

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