charmatrix_fromBlocks — Mathlib

English

In a 2×2 matrix context, hyperbolicity is preserved under conjugation by an invertible matrix: IsHyperbolic(g^{-1} m g) holds iff IsHyperbolic(m).

Русский

При конъюгации обратимой матрицей гиперболичность сохраняется: IsHyperbolic(g^{-1} m g) эквивалентно IsHyperbolic(m).

LaTeX

$$$\\mathrm{IsHyperbolic}(g^{-1} m g) \\iff \\mathrm{IsHyperbolic}(m)$$$

Lean4

theorem charmatrix_fromBlocks :
    charmatrix (fromBlocks M₁₁ M₁₂ M₂₁ M₂₂) = fromBlocks (charmatrix M₁₁) (-M₁₂.map C) (-M₂₁.map C) (charmatrix M₂₂) :=
  by
  simp only [charmatrix]
  ext (i | i) (j | j) : 2 <;>
    simp [diagonal]
      -- TODO: importing block triangular here is somewhat expensive, if more lemmas about it are added
      -- to this file, it may be worth extracting things out to Charpoly/Block.lean

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