In a paper I have read, the authors reformulated the following LMI
$$X_{t} \succeq \left[\begin{array}{cc}\alpha_{i}\left(B_{t} U+C_{t}\right)^{\top} E_{i}\left(B_{t} U+C_{t}\right) & \left( B_{t} U+C_{t}\right)^{\top} e_{i} \alpha_{i} \\\\ \alpha_{i} e_{i}^{\top}\left( B_{t} U+C_{t}\right) & e_{i}^{0} \alpha_{i}-\beta_{t}\end{array}\right]$$
into two LMIs
$$X_{t} - \left[\begin{array}{cc}\alpha_{i}P_{t}^{i} & \left( B_{t} U+C_{t}\right)^{\top} e_{i} \alpha_{i} \\\\ \alpha_{i} e_{i}^{\top}\left( B_{t} U+C_{t}\right) & e_{i}^{0} \alpha_{i}-\beta_{t}\end{array}\right] \succeq 0$$
and
$$\left[\begin{array}{cc}P_{t}^{i} & \left(B_{t} U+C_{t}\right)^{\top} E_{i}^{1 / 2} \\\\ E_{i}^{1 / 2}\left(B_{t} U+C_{t}\right) & \mathbb{I}_{n}\end{array}\right] \succeq 0 $$
where $\alpha_i$ and $e_i^0$ are scalar and $e_i$ is vector. The decision variables are $X_{t} \in S_{+}^{N d+2}$ and $P_{t}^{i} \in S_{+}^{N d+1}$.
I know the second LMI could be reformulated into $$P_{t}^{i} - \left(B_{t} U + C_{t}\right)^{\top} E_{i} \left(B_{t} U + C_{t}\right) \succeq 0$$ according to Schur complements. Just wondering why the two LMIs are equivalent to the first LMI and why the reformulation is necessary.
Any hint would be appreciated!
The first semidefinite constraint is not an LMI as they have quadratic term in the top-left block (assuming $U$, $B_t$ or $C_t$ is a decision variable). However, it is convex in the standard form $A-B^TCB$ (generic variable names here) so they introduce an intermediate variable to upper bound the nonlinear part, and then apply a Schur complement to arrive at two LMIs. An alternative could be to apply a Schur complement directly on the first semidefinite constraint arriving at one larger LMI but without any extra variable.