The relative gain array of a real invertible matrix 
is 
,
where 
denotes the Hadamard product. The inverse relative
gain array is its matrix inverse. For a real symmetric
positive definite matrix 
, it is
 |
It has unit row and column sums, so it is doubly stochastic exactly when it is entrywise nonnegative.
Wang (2026) proved that 
is entrywise nonnegative for every real symmetric positive
definite matrix 
of order at most 6, resolving Uhlmann's inverse relative gain
array conjecture through the claimed range. A rational order-7 counterexample shows
that the dimension bound is sharp.
Wang (2026) reports that GPT-6 Astra assisted with brainstorming, mathematical development, and manuscript drafting.
