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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?

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    $\begingroup$ Why do you say there is no nonzero $ v $ such that $ Mv = 0 $? $\endgroup$ – user571438 Jul 1 '18 at 17:17
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    $\begingroup$ 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. $\endgroup$ – Marlo Jul 1 '18 at 17:22
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All nonzero vectors are eigenvectors, since all vectors $v$ satisfy $Mv=0v$ if $M$ is the zero matrix.

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  • $\begingroup$ Thanks @Hans. I can't believe how I have not seen this :) $\endgroup$ – Marlo Jul 1 '18 at 17:24
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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.

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