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

The rank of the $ n \times n$ $ (A - \lambda I)$ $ n - J(\lambda )$ is an eigenvalue occurring in $ J(\lambda ) \geqq 0$. In our principal theorem we derive an analogous expression for the rank of $ (A - \lambda B)$ $ m \times n$ 0,\lambda$ --> $ J(\lambda ) > 0,\lambda $ $ (A - \lambda I)$ $ m \times n$ $ (A - \lambda B)$ $ P(\lambda ,A,B)$ $ (A - \lambda B)$ relative to <IMG WIDTH="22" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img17.gif" ALT="$ B$"> and justify that name.