DibyashanuPati/PhysLean
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

clean mathematics

Browse source
This commit is contained in:
Pietro Monticone 2025-01-14 00:11:17 +01:00
parent fa425ef606
commit d911b3b0f9
5 changed files with 9 additions and 9 deletions
Download patch file Download diff file Expand all files Collapse all files

6
HepLean/Mathematics/SchurTriangulation.lean
View file

@ -17,7 +17,7 @@ closed field, e.g., `ℂ`, is unitarily similar to an upper triangular matrix.
be decomposed as `A = U * T * star U` where `U` is unitary and `T` is upper triangular.
- `Matrix.schurTriangulationUnitary` : the unitary matrix `U` as previously stated.
- `Matrix.schurTriangulation` : the upper triangular matrix `T` as previously stated.
- Some auxilary definitions are not meant to be used directly, but
- Some auxiliary definitions are not meant to be used directly, but
`LinearMap.SchurTriangulationAux.of` contains the main algorithm for the triangulation procedure.
-/
@ -124,8 +124,8 @@ end
Given a linear endomorphism `f` on a non-trivial finite-dimensional vector space `E` over an
algebraically closed field `𝕜`, one can always pick an eigenvalue `μ` of `f` whose corresponding
eigenspace `V` is non-trivial. Given that `E` is also an inner product space, let `bV` and `bW` be
othonormal bases for `V` and `Vᗮ` respectively. Then, the collection of vectors in `bV` and `bW`
forms an othornomal basis `bE` for `E`, as the direct sum of `V` and `Vᗮ` is an internal
orthonormal bases for `V` and `Vᗮ` respectively. Then, the collection of vectors in `bV` and `bW`
forms an orthonormal basis `bE` for `E`, as the direct sum of `V` and `Vᗮ` is an internal
decomposition of `E`. The matrix representation of `f` with respect to `bE` satisfies
$$
\sideset{_\mathrm{bE}}{_\mathrm{bE}}{[f]} =