Eigendecomposition of a block matrix of the form $\begin{pmatrix} A_{11} & A_{12} \\ A_{12}^T & 0 \end{pmatrix}$

Question

Consider a matrix $A = \begin{pmatrix} A_{11} & A_{12} \\ A_{12}^T & 0 \end{pmatrix}$ where $A_{11}$ is symmetric. The eigendecomposition of $A_{11}$ is $A_{11} = U_1S_1U_1^T$ and the one of of $A$ is $A = USU^T$. Then set $X_1 = U_1|S_1|^{1/2}$ such that $ A_{11} = X_1 I_{p_1q_1} X_1^T $ where $(p_1,q_1)$ is the signature of $A_{11}$ and similarly $X = U|S|^{1/2}$ such that $ A = X I_{pq} X^T $ where $(p,q)$ is the signature of $A$. Assume $A_{11}$ is of dimension $n \times n$ and $ A_{12}$ is of dimensions $n \times p$. I want to find a relationship between the first $n$ rows of $X$ and $X_1$. Any hints on how I can do this?