Plain-language statement
The determinant of the exponential of a matrix is the exponential of its trace. This is also known as Lie's trace formula.
Exact Lean statement
theorem det_exp {๐ m : Type*} [RCLike ๐] [IsAlgClosed ๐] [Fintype m] [LinearOrder m]
(A : Matrix m m ๐) :
(NormedSpace.exp A).det = NormedSpace.exp A.traceFormal artifact
Lean source
theorem det_exp {๐ m : Type*} [RCLike ๐] [IsAlgClosed ๐] [Fintype m] [LinearOrder m] (A : Matrix m m ๐) : (NormedSpace.exp A).det = NormedSpace.exp A.trace := by let U := A.schurTriangulationUnitary let T := A.schurTriangulation have h_prop : T.val.IsUpperTriangular := T.property have h_conj : A = U * T * star U := schur_triangulation A have h_trace_invariant : A.trace = T.val.trace := by rw [h_conj, Unitary.coe_star, trace_unitary_conj] have h_det_invariant : (NormedSpace.exp A).det = (NormedSpace.exp T.val).det := by rw [h_conj, Unitary.coe_star, det_exp_unitary_conj] have h_triangular_case : (NormedSpace.exp T.val).det = NormedSpace.exp T.val.trace := det_exp_of_blockTriangular_id h_prop rw [h_det_invariant, h_triangular_case, h_trace_invariant]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Mathematics/DataStructures/Matrix/LieTrace.lean:197-210
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