Theorems · Theorem · linear algebra

LDL.diag_eq_lowerInv_conj

∀ {𝕜 : Type u_1} [inst : RCLike 𝕜] {n : Type u_2} [inst_1 : LinearOrder n] [inst_2 : WellFoundedLT n]
  [inst_3 : LocallyFiniteOrderBot n] {S : Matrix n n 𝕜} [inst_4 : Fintype n] (hS : S.PosDef),
  LDL.diag hS = LDL.lowerInv hS * S * (LDL.lowerInv hS).conjTranspose

Inverse statement of LDL decomposition: we can conjugate a positive definite matrix by some lower triangular matrix and get a diagonal matrix.

Defined in

Mathlib.Analysis.Matrix.LDL

Cited by

1 results in Mathlib

Foundations

Depth 229 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeLinearOrderWellFoundedLTLocallyFiniteOrderBotFintype