Eigenvalues are characteristic polynomial roots. #
In fields we show that:
Matrix.det_eq_prod_roots_charpoly_of_splits: the determinant (in the field of the matrix) is the product of the roots of the characteristic polynomial if the polynomial splits in the field of the matrix.Matrix.trace_eq_sum_roots_charpoly_of_splits: the trace is the sum of the roots of the characteristic polynomial if the polynomial splits in the field of the matrix.
In an algebraically closed field we show that:
Matrix.det_eq_prod_roots_charpoly: the determinant is the product of the roots of the characteristic polynomial.Matrix.trace_eq_sum_roots_charpoly: the trace is the sum of the roots of the characteristic polynomial.
Note that over other fields such as ℝ, these results can be used by using
A.map (algebraMap ℝ ℂ) as the matrix, and then applying RingHom.map_det.
The two lemmas Matrix.det_eq_prod_roots_charpoly and Matrix.trace_eq_sum_roots_charpoly are more
commonly stated as trace is the sum of eigenvalues and determinant is the product of eigenvalues.
Mathlib has already defined eigenvalues in LinearAlgebra.Eigenspace as the roots of the minimal
polynomial of a linear endomorphism. These do not have correct multiplicity and cannot be used in
the theorems above. Hence we express these theorems in terms of the roots of the characteristic
polynomial directly.
TODO #
The proofs of det_eq_prod_roots_charpoly_of_splits and
trace_eq_sum_roots_charpoly_of_splits closely resemble
norm_gen_eq_prod_roots and trace_gen_eq_sum_roots respectively, but the
dependencies are not general enough to unify them. We should refactor
Polynomial.coeff_zero_eq_prod_roots_of_monic_of_split and
Polynomial.nextCoeff_eq_neg_sum_roots_of_monic_of_splits to assume splitting over an
arbitrary map.