# 2022-14 The number of eigenvalues of a symmetric matrix

For a positive integer $$n$$, let $$B$$ and $$C$$ be real-valued $$n$$ by $$n$$ matrices and $$O$$ be the $$n$$ by $$n$$ zero matrix. Assume further that $$B$$ is invertible and $$C$$ is symmetric. Define $A := \begin{pmatrix} O & B \\ B^T & C \end{pmatrix}.$ What is the possible number of positive eigenvalues for $$A$$?

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