Equivalences.

Suppose $X$ and $Y$ are nonsingular matrices. Let $\hat{A}= Y^*AX$ and $\hat{B}= Y^*BX$. We say that the pencil $\hat{A}- \lambda \hat{B}$ is equivalent to $A - \lambda B$ and that $X$ and $Y$ are equivalence transformations. $\hat{A}- \lambda \hat{B}$ and $A - \lambda B$ have the same eigenvalues. If $\cal X$ ($\cal Y$) is a right (left) reducing subspace of $A - \lambda B$, then $\hat{\cal X}=X^{-1} {\cal X}$ ( $\hat{\cal Y} = Y^{*}{\cal Y}$) is a right (left) reducing subspace of $\hat{A}- \lambda \hat{B}$.