Consider dynamical system $\dot{x} = Ax$ where $x \in \mathbb{R}^n$?
What can we say about eigenvalues of $A$ when $\dim\operatorname{Null}(A)=2$? Can we say that $\lambda = 0$ is an eigenvalue with multiplicity two?
2026-03-29 10:48:12.1774781292
What can we say about eigenvalues of $A$ when $\dim\operatorname{Null}(A)=2$?
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Sure, because the nullspace of $A$ has two linearly independent vectors $v_1,v_2 \ne 0$ so that $Av_1 = Av_2 = 0 = 0v_1 = 0v_2$. Therefore, $\lambda = 0$ is an eigenvalue with geometric multiplicity two, and algebraic multiplicity $\ge 2$.