Conditions for a matrix to be invertible Let $n \geq m$ and let $C$ be a $n \times m$ full rank matrix, that is $rank(C) =m$. Considering that $D$ is a diagonal positive semidefinite matrix, under which conditions is the $ m \times m$ matrix $X=C^\mathsf{T}DC$ invertible? Is it enough to have $rank(D) \geq m$?

If $D$ is positive definit, and $C$ has full rank ($C\in \mathbb R^{n,m}$, $n\ge m$, $rank(C)=m$), then $C^TDC$ is positive definite, hence invertible: Take $x\ne 0$ then
$$
x^TC^TDCx  = (Cx)^TD(Cx) >0
$$
as $D$ is positive definite and $Cx\ne 0$.
