Schur triangulation
Matrix.schur_triangulation
Plain-language statement
Schur triangulation, Schur decomposition for matrices over an algebraically closed field. In particular, a complex matrix can be converted to upper-triangular form by a change of basis. In other words, any complex matrix is unitarily similar to an upper triangular matrix.
Exact Lean statement
lemma schur_triangulation :
A = A.schurTriangulationUnitary * A.schurTriangulation * star A.schurTriangulationUnitaryFormal artifact
Lean source
lemma schur_triangulation : A = A.schurTriangulationUnitary * A.schurTriangulation * star A.schurTriangulationUnitary := let U := A.schurTriangulationUnitary have h : U * A.schurTriangulation.val = A * U := let b := A.schurTriangulationBasis.toBasis let c := (EuclideanSpace.basisFun n ๐).toBasis calc c.toMatrix b * LinearMap.toMatrix b b (toEuclideanLin A) _ = LinearMap.toMatrix c c (toEuclideanLin A) * c.toMatrix b := by simp _ = LinearMap.toMatrix c c (toLin c c A) * U := rfl _ = A * U := by simp calc A _ = A * U * star U := by simp [mul_assoc] _ = U * A.schurTriangulation * star U := by rw [โ h]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Mathematics/SchurTriangulation.lean:278-290
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