The Minimality of Determinantal Varieties

The determinantal variety ${\mathrm{\Sigma }}_{pq}$ is defined to be the set of all $p\times q$ real matrices with $p\ge q$ whose ranks are strictly smaller than $q$. It is proved that ${\mathrm{\Sigma }}_{pq}$ is a minimal cone in ${\mathbb{R}}^{pq}$ and all its strata are regular minimal submanifolds.