Theorems · Theorem · combinatorics

Matrix.IsHadamard.of_mul_conjTranspose

∀ {n : Type u_2} {R : Type u_3} [inst : Fintype n] [inst_1 : DecidableEq n] [inst_2 : CommRing R] [inst_3 : StarRing R]
  {A : Matrix n n R},
  (∀ (i j : n), A i j ∈ unitary R) →
    A * A.conjTranspose = ↑(Fintype.card n) • 1 → IsRegular ↑(Fintype.card n) → A.IsHadamard

Build a Hadamard matrix from the one-sided row-orthogonality condition, provided the order is regular in R. This is the matrix form of [Theorem 2.3.6][deLauneyFlannery2011].

Defined in

Mathlib.LinearAlgebra.Matrix.HadamardMatrix

Cited by

1 results in Mathlib

Foundations

Depth 100 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FintypeDecidableEqCommRingStarRing