Section CharPoly.
Variables (R : ringType) (n : nat) (A : 'M[R]_n).
Implicit Types p q : {poly R}.
Definition char_poly_mx := 'X%:M - map_mx (@polyC R) A.
Definition char_poly := \det char_poly_mx.
Let diagA := [seq A i i | i <- index_enum _ & true].
Let size_diagA : size diagA = n.
End CharPoly.
Prenex Implicits char_poly_mx char_poly.

Lemma char_poly_trig {R : comRingType} n (A : 'M[R]_n) : is_trig_mx A ->
char_poly A = \prod_(i < n) ('X - (A i i)%:P).