English
For any A,B,C, trace is invariant under algebra equivalences: trace_A B x equals trace_A C (some transformed x) via the equivalence.
Русский
Для любых A,B,C след сохраняется под алгебраическими эквивалентностями: след A B x равен следу через преобразование под эквивалентностью.
LaTeX
$$Eq (LinearMap.coe (Algebra.trace A C) x) (EquivLike.coe e_symm (LinearMap.coe (Algebra.trace A B) x))$$
Lean4
/-- Given an `A`-algebra `B` and `b`, a `κ`-indexed family of elements of `B`, we define
`traceMatrix A b` as the matrix whose `(i j)`-th element is the trace of `b i * b j`. -/
noncomputable def traceMatrix (b : κ → B) : Matrix κ κ A :=
of fun i j =>
traceForm A B (b i)
(b j)
-- TODO: set as an equation lemma for `traceMatrix`, see https://github.com/leanprover-community/mathlib4/pull/3024
