Consider the generalized eigenvalue problem (GEVP)
$\begin{equation} \boldsymbol{A} \boldsymbol{v} = \lambda \boldsymbol{B} \boldsymbol{v} \tag{1}\label{1} \end{equation} $
where $\boldsymbol{A}, \boldsymbol{B} \in \mathbb{R}^{n \times n}$ are both singular.
(1) Is the algebraic multiplicity of $\lambda = 0$ for the GEVP always the same as the algebraic multiplicity of $\lambda = 0$ for $\boldsymbol{A}$? And is the proof as simple as noting that for $\lambda = 0$ $\eqref{1}$ is
$\begin{equation} \boldsymbol{A} \boldsymbol{v} = 0. \tag{2}\label{2} \end{equation} $
(2) Consider that the eigenvalues of $\boldsymbol{A}$ and $\boldsymbol{B}$ are known. I understand the count of positive, negative and null eigenvalues of the GEVP can be known, in the case where $\boldsymbol{B}$ is positive definitive [1], but can some insight be gained in this case where both $\boldsymbol{A}$ and $\boldsymbol{B}$ are singular?
Thank you!