I have a symmetric positive definite (SPD) matrix $A\in\mathbb{R}^{n\times n}$ and a full-rank matrix $B\in\mathbb{R}^{m\times n}$.
I need to show that the pre-conditioned matrix $\begin{bmatrix} A & 0 \\ 0 & BA^{-1}B^T \end{bmatrix}^{-1}\begin{bmatrix} A & B^T \\ B & 0 \end{bmatrix}$ has only three eigenvalues, $1, (1\pm\sqrt{5})/2,$ and I need to find their multiplicity.
I do not understand why this problem is interesting or how it was created. Why does the golden ratio appear at all here? I also do not know how to get such specific eigenvalues from such a seemingly-general matrix.