- #1
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If [tex]X\in\textrm{End}(\mathbb{R}^n)[/tex] is some arbitrary nxn-matrix, is it true that
[tex]
X^2 = 0\quad\implies\quad \textrm{Tr}(X)=0?
[/tex]
[tex]
X^2 = 0\quad\implies\quad \textrm{Tr}(X)=0?
[/tex]
Sounds plausible. Doesn't it follow immediately from putting X into a normal form, or from computing its generalized eigenvalues?If [tex]X\in\textrm{End}(\mathbb{R}^n)[/tex] is some arbitrary nxn-matrix, is it true that
[tex]
X^2 = 0\quad\implies\quad \textrm{Tr}(X)=0?
[/tex]
If X^2 is zero then either X is zero, or X^2 is the minimal poly of X, and in particular divides its characteristic polynomial, which implies that the char poly has 0 constant term, but the constant term is +/- the trace of X. No need to invoke a choice of basis.