Theorems · Theorem · linear algebra

Matrix.det_sum_smul_le

∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [Nontrivial R] [inst_2 : CommRing S] [inst_3 : LinearOrder S]
  [IsStrictOrderedRing S] {n : Type u_3} [inst_5 : Fintype n] [inst_6 : DecidableEq n] {ι : Type u_4} (s : Finset ι)
  {c : ι → R} {A : ι → Matrix n n R} {abv : AbsoluteValue R S} {x : S},
  (∀ (k : ι) (i j : n), abv (A k i j) ≤ x) →
    ∀ {y : S},
      (∀ (k : ι), abv (c k) ≤ y) →
        abv (∑ k ∈ s, c k • A k).det ≤ (Fintype.card n).factorial • (s.card • y * x) ^ Fintype.card n

Defined in

Mathlib.LinearAlgebra.Matrix.AbsoluteValue

Cited by

1 results in Mathlib

Foundations

Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingNontrivialCommRingLinearOrderIsStrictOrderedRingFintypeDecidableEq