If an upper Hessenberg matrix has a zero on the subdiagonal, the problem must be split into two eigenvalue problems. Assume a split has occurred and the matrix has the form where 11 is   22 is  and...

If an upper Hessenberg matrix has a zero on the subdiagonal, the problem must be split into two eigenvalue problems. Assume a split has occurred and the matrix has the form

where

₁₁
is

₂₂
is
 and both are upper Hessenberg. Show that the eigenvalues of
 are those of

₁₁
and

_22.
For the sake of simplicity, assume that

₁₁
and

₂₂
have no common eigenvalues and that if
 is an eigenvalue of

₂₂
then the matrix

₁₁
 is nonsingular.