Originally Posted by

**wilbursmith** Donīt know if the descriptive title is misleading but here goes.

Show that for every $\displaystyle m/n$matrix $\displaystyle A$ with a range $\displaystyle r$ there exists a matrix $\displaystyle AA^T$ which is symmetric and has a range $\displaystyle r$.

This is what I have:

The matrix $\displaystyle AA^T$ is a square $\displaystyle m/m$ matrix and also symmetric because $\displaystyle (AA^T)^T=A^{TT}A^T=AA^T$.

The part with range is still confusing. Now this should be true

$\displaystyle dimR(A)=dimR(A^T)=r$

but how should I pursue $\displaystyle dimR(AA^T)=r$?