Theorems · Theorem · linear algebra

Matrix.IsUnit.posDef_star_right_conjugate_iff

∀ {n : Type u_2} {R : Type u_3} [inst : Ring R] [inst_1 : PartialOrder R] [inst_2 : StarRing R] [inst_3 : Fintype n]
  [inst_4 : DecidableEq n] {x U : Matrix n n R}, IsUnit U → ((U * x * star U).PosDef ↔ x.PosDef)

For an invertible matrix U, U * x * star U is positive definite iff x is. This works on any ⋆-ring with a partial order. See IsUnit.isStrictlyPositive_star_right_conjugate_iff for a similar statement for star-ordered rings. For matrices, positive definiteness is equivalent to strict positivity when the underlying field is ℝ or ℂ (see Matrix.isStrictlyPositive_iff_posDef).

Defined in

Mathlib.LinearAlgebra.Matrix.PosDef

Cited by

1 results in Mathlib

Foundations

Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingPartialOrderStarRingFintypeDecidableEq