An exact Francis step terminates with unitary similarity and its stated status

If $H$ is upper Hessenberg, then $H{e}_1$ is supported in rows $1,2$; applying $H$ once more makes $H^2{e}_1$ supported in rows $1,2,3$. Thus the defined $p(H)e_1$ is supported in its first $r+1$ entries for either shift descriptor.

If the initial vector is zero, the algorithm returns the identity step. Otherwise [F2] supplies $P_0$. Each later active vector either is zero, in which case its identity branch is defined, or [F2] supplies $P_j$. The loop has at most $\max (n-2,0)$ iterations, so the step terminates. In the ordinary real branch [F3] identifies the trace/determinant polynomial with the conjugate-pair shifts when the trailing block has negative discriminant.

Every update is the exact similarity $H_{j+1}=P_j^{\ast }{H}_j{P}_j$. Hence the product $Q=P_0{P}_1\cdots P_j$ gives $H_{j+1}=Q^{\ast }HQ$ by induction on the finite update list; every factor is unitary by [F2].

In the nonzero-vector branch, [F4] proves that the final matrix is upper Hessenberg and $Q{e}_1$ is proportional to $p(H)e_1$, so the exact checks return completed. The zero-vector branch already returned its explicit status. Hence the invariant-failure branch is unreachable for valid exact inputs, and every returned status and similarity factor is well-defined.