A generalized eigenvalue problem

Question

The generalized eigenvalue problem likes this:

$\begin{pmatrix} 0 & C_{12}\\ C_{21} & 0 \end{pmatrix} \begin{pmatrix} \xi_1\\ \xi_2 \end{pmatrix}=\rho\begin{pmatrix} C_{11} & 0\\ 0 & C_{22} \end{pmatrix}\begin{pmatrix} \xi_1\\ \xi_2 \end{pmatrix}$,

where $\xi_1$ and $\xi_2$ have $p_1$ and $p_2$ dimensions, respectively. This problem has $p_1+p_2$ eigenvalues: $\{\rho_1, -\rho_1, ..., \rho_p, -\rho_p, 0,...., 0\}$, where $p=min\{p_1, p_2\}$.

Would somebody help me to prove both $\rho_i$ and $-\rho_i$ are the eigenvalues and provide the ranges for them? Also prove $p=min\{p_1, p_2\}$? Thanks in advance:)

Answer 1

You consider the GE problem $Ax=\lambda Bx$. Then the eigenvalues are the roots in $\lambda$ of $\det(A-\lambda B)=0$.

Here $\det(A-\lambda B)=Q(\lambda)=\det(\begin{pmatrix}-\lambda C_{11}&C_{12}\\C_{21}&-\lambda C_{22}\end{pmatrix}$.

$Q$ is a polynomial of degree $p_1+p_2$ that has same parity as $p_1+p_2$. Since $rank(A)\leq 2p$, there are at most $2p$ non-zero roots of $Q$ and the eigenvalues are in the form $\rho_1,-\rho_1,\cdots,\rho_p,-\rho_p$.