For any $m \times n$ matrix $A$, the matrices $A^{t}A$ and $AA^{t}$ are positive semi- definite. > For any $m \times n$ matrix $A$, then $A^{t}A$ and $AA^{t}$ are positive semi-definite. How do I prove the theorem using the definition of positive semi-definiteness? I know the definition, but the definition is not helping me. Please give me hints. How do I work out the given problem?

As stated the result is not true as you can see by taking the null matrix. But for all matrices $A$ the products $AA^t$ and $A^tA$ are positive _semidefinite_.

Hint: Just use the definition and use the fact that $(AB)^t=B^tA^t$ and that vectors can be seen as matrices, too.