Consider a block, symmetric matrix $$ \begin{pmatrix} 0 & -A^T \\ A & C \end{pmatrix} $$ where $A$ and $C$ are two real positive definite matrices. What is the condition number of that matrix? Is it related to the condition number of $A^T C^{-1} A$?

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Consider a block, symmetric matrix $$ \begin{pmatrix} 0 & -A^T \\ A & C \end{pmatrix} $$ where $A$ and $C$ are two real positive definite matrices. What is the condition number of that matrix? Is it related to the condition number of $A^T C^{-1} A$?