Eigenvectors and eigenvalues of the zero matrix

Question

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: \begin{equation} Mv=\lambda v=0 \end{equation} Hence no eigenvectors and no eigenvalues? Or would one say that 0 is an eigenvalue without a corresponding eigenvector?

Answer 1

All nonzero vectors are eigenvectors, since all vectors $v$ satisfy $Mv=0v$ if $M$ is the zero matrix.

Answer 2

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.