**In system theory, we often encounter Semi-Definite Programs (SDPs) with Linear Matrix Inequality (LMI) constraints, such as those presented in this paper. I have introduced new variables based on the original decision variables and reformulated the LMIs using these new variables. Additionally, I have defined a similar SDP and, through simulation, identified relationships between their optimal solutions. I am now seeking mathematical proofs to substantiate these relationships. Further details will be provided below, and I would greatly appreciate any assistance in addressing my inquiry.**

Consider following LMIs, where H \in \mathbb{R}^{(n+m) \times (n+m)} and W \in \mathbb{R}^{n \times n} are decision variables (0 <m \leq n) and we have:

Also, D \in \mathbb{R}^{n \times (n+m)} and C \succeq 0 are constant matrices. Additionally, in all equations, 0 represents zero matrices of appropriate dimensions.

Suppose K \in \mathbb{R}^{n \times j} is a constant matrix with rank n and j = n + n \cdot m . The matrix T \in \mathbb{R}^{(n+m) \times (j+m)} is constructed as:

Suppose there exists a matrix S \in \mathbb{R}^{j \times (j+m)} such that D \cdot T = K \cdot S .

The new variable X \in \mathbb{R}^{(j+m) \times (j+m)} is defined as:

Thus, the following relationships are established between the submatrices of H and X :

If we multiply the above LMIs from the left and right by T^T and T , respectively, we obtain:

Selecting the new variable V as V = K^T W K , we construct two SDPs as follows.

**First SDP:**

**Second SDP:**

**My Question:**

Let H^* and W^* be the optimal solutions of the first SDP, and X^* and V^* the optimal solution of the second SDP.

We define Z^* as follows:

I seek to prove the following equalities:

- V^*= K^T W^* K
- ({X}_{22}^*)^{-1} \cdot ({X}_{12}^*)^T = ({Z}_{22}^*)^{-1} \cdot ({Z}_{12}^*)^T

note: Using Schur complement property, the equivalent of second LMIs substituted

**My Attempt:**

I have tested numerous examples with n \leq 6 and m \leq n using randomly generated K and D matrices, and in all cases, the equalities hold. However, I am uncertain how to establish a mathematical proof for these results.