Triangularize matrices over an algebraically closed field - #410
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Adds RealRooted/Mathlib/LinearAlgebra/Matrix/Triangularize.lean: - `Matrix.exists_mulVec_eq_smul`: a matrix over an algebraically closed field has an eigenvector (charpoly root -> singular pencil -> kernel vector); - `Matrix.exists_unitsConj_blockTriangular`: every square matrix over an algebraically closed field is conjugate to an upper triangular matrix, the triangular shape witnessed by an injective height function `b : m -> N`, so no linear order on the index type is required. The induction conjugates by `1.updateCol i0 v` (invertible since its determinant is `v i0`, via `cramer_one`) to clear the eigencolumn, then splits the index type with `Equiv.sumCompl` and recurses on the complement block; all block bookkeeping is `Matrix.fromBlocks` algebra plus `submatrix_mul_equiv`. This is the missing keystone for the compound eigenvalue theorem (#405 item 3) and, downstream, Gantmacher-Krein (#394). Mathlib currently has no matrix-level triangularization; the file mirrors the Mathlib tree for upstreaming. Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
This was referenced Aug 21, 2026
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The missing keystone for #405 item 3 (and, downstream, #394).
New file
RealRooted/Mathlib/LinearAlgebra/Matrix/Triangularize.lean:Matrix.exists_mulVec_eq_smul: a matrix over an algebraically closed fieldhas an eigenvector — charpoly root → singular pencil (
spectrum.mem_iff+isUnit_iff_isUnit_det) → kernel vector (exists_mulVec_eq_zero_iff);Matrix.exists_unitsConj_blockTriangular: every square matrix over analgebraically closed field is conjugate to an upper triangular matrix,
with the triangular shape witnessed by an injective height function
b : m → ℕ— index-type agnostic, no linear order needed.Proof: classical induction on
Fintype.card. Conjugating by1.updateCol i₀ v(invertible: its determinant isv i₀viacramer_one)clears the eigencolumn;
Equiv.sumComplsplits the index type; the complementblock recurses; all bookkeeping is
fromBlocksalgebra +submatrix_mul_equiv.Mathlib has no matrix-level triangularization (
Eigenspace/Triangularizablestops at flags of invariant subspaces), so the file mirrors the Mathlib tree
for upstreaming.
Together with #409 (
charpoly_compound_of_blockTriangular,charpoly_compound_conj) this makes #405 item 3 a pure assembly step:triangularize, take compounds, read off the diagonal.
Sorry-free, no warnings.
🤖 Generated with Claude Code