Reducing the rank of $(A-\lambda B)$

The rank of the $n \times n$ matrix $(A - \lambda I)$ is $n - J(\lambda )$ when $\lambda$ is an eigenvalue occurring in $J(\lambda ) \geqq 0$ Jordan blocks of the Jordan normal form of $A$. In our principal theorem we derive an analogous expression for the rank of $(A - \lambda B)$ for general, $m \times n$, matrices. When $J(\lambda ) > 0,\lambda$ is a rank-reducing number of $(A - \lambda I)$. We show how the rank-reducing properties of eigenvalues can be extended to $m \times n$ matrix expressions $(A - \lambda B)$. In particular we give a constructive way of deriving a polynomial $P(\lambda ,A,B)$ whose roots are the only rank-reducing numbers of $(A - \lambda B)$. We name this polynomial the characteristic polynomial of $A$ relative to $B$ and justify that name.