Can $X - Y A^\dagger Y^T\succ0$ be written as an LMI where $A^\dagger$ is a pseudoinverse?
Solution 1:
As mentioned in the comments, in the general case where $A\succeq0$ may not be invertible there is an orthogonality condition that reads as follows: \begin{align} X-YA^\dagger Y^T\succeq0 \ \ \text{and}\ \ Y(I - A A^\dagger)=0 \iff \begin{pmatrix} X&Y\\ Y^T &A\end{pmatrix}\succeq0. \end{align}