∀ {n : Type u_2} {R : Type u_4} [inst : Fintype n] [inst_1 : PartialOrder R] [inst_2 : NonUnitalRing R]
  [inst_3 : StarRing R] [StarOrderedRing R] [NoZeroDivisors R] {m : Type u_5} {A : Matrix m n R},
  (A * A.conjTranspose = 0) = (A = 0)