Section KernelLemmas.
Variable K : fieldType.
(* convertible to kermx (horner_mx g p) when n = n.+1 *)
Definition kermxpoly n (g : 'M_n) (p : {poly K}) : 'M_n :=
kermx ((if n is n.+1 then horner_mx^~ p : 'M_n.+1 -> 'M_n.+1 else \0) g).
Definition geigenspace n (g : 'M_n) a := kermxpoly g (('X - a%:P) ^+ n).

Lemma eigenspace_sub_geigen n (g : 'M_n) a :
(eigenspace g a <= geigenspace g a)%MS.