I have an expression of the form:
$ACA′$
where C is an invertible, symmetric and positive definite matrix. I'm trying to figure out if the expression above is invertible (or what additional properties C would need to have). The C matrix is $n$x$n$ and the A matrix is $k$x$n$ ($k<n$). Every row of A has exactly one entry =1 and one entry =-1.
Any help, especially something pointing to a theorem, would be greatly appreciated (does A need to be of full row rank?).
The ones and minus ones are not relevant since they occur as rows in $A$ but as columns in $A^T.$
I am going to write $B = A^T.$ As $C$ is symmetric positive definite, I am also going to write, for example by a Cholesky decomposition, as $$ C = H^T H. $$ So, your expression becomes $$ B^T H^T H B, $$ where $B$ has maximal rank, that being $k < n.$
Finally, I take a column vector $v \neq 0$ with $k$ entries. As the row rank and column rank of $B$ are the same, we find $Bv \neq 0. $ As $H$ is nonsingular, $HBv \neq 0.$ The dot product $(HBv) \cdot (HBv) \neq 0.$ So, $$ v^T B^T H^T H B v \neq 0. $$ In particular, $$ B^T H^T H B v \neq 0. $$ So, $$ B^T H^T H B $$ is square and nonsingular and therefore invertible; it is also symmetric positive definite.
If your matrix $A$ has less than maximal rank, you get a singular result instead.