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Let $A$ be an $m\times n$ matrix. Prove by explicit example that zero can be an eigenvalue of one of the matrices $A^{T}A$, $AA^{T}$ and not of the other.

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To add to the answer already there: $A$ has this property iff $A$ is non-square and either $A$ or $A^T$ has a nullspace of ${0}$. –  Omnomnomnom Jun 9 '13 at 20:49

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Let $A=[1~ 1]$. Then $A^TA$ is a degenerate $2\times 2$ matrix, while $AA^T=[2]$.

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