# Eigenvectors and eigenvalues of the zero matrix

Consider the zero matrix $M=o$.

Is it correct to say that $M$ then has no eigenvalues and eigenvectors? A natural guess for a candidate would be $\lambda=0$. It solves the characteristic equation $\det (M-\lambda \mathbb I)=0$. But there is no associated eigenvector, that is a nonzero vector $v$ such that: $$Mv=\lambda v=0$$ Hence no eigenvectors and no eigenvalues? Or would one say that 0 is an eigenvalue without a corresponding eigenvector?

• Why do you say there is no nonzero $v$ such that $Mv = 0$? Jul 1, 2018 at 17:17
• quite right. I'm not seeing the forest for the trees. Eigenvectors everywhere :) I knew this question was embarrassing, but fortunately I'm not afraid of embarrassment. Jul 1, 2018 at 17:22

All nonzero vectors are eigenvectors, since all vectors $v$ satisfy $Mv=0v$ if $M$ is the zero matrix.
• I would like to add that therefore the only eigenvalue is $0$. Jul 8, 2021 at 15:25
The equation $Mv=\lambda v$ is satisfied by $\lambda =0$ and every non-zero vector.
Thus you have an eigenvalue of $0$ and all non-zero vectors as your eigenvectors.