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