Can $X - Y A^\dagger Y^T\succ0$ be written as an LMI where $A^\dagger$ is a pseudoinverse?

Question

I have the constraint

\begin{align} X - Y A^\dagger Y^T\succ0, \end{align}

where $A^\dagger$ is the pseudoinverse of $A\succeq0$. Can we still use the Schur complement to write the constraint as an LMI?

Explicitly, can we show something like:

\begin{align} X - Y A^\dagger Y^T\succeq0 &\iff \begin{pmatrix}X&Y\\Y^T&A\end{pmatrix}\succeq0 \end{align}

Answer: It is true only when $A$ is invertible, please see answer below on the general case.

Answer 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}