The inverse relative gain array of a real invertible matrix 
is the matrix inverse of its relative gain array,
when that inverse exists. The relative gain array is 
, where 
denotes the Hadamard
product. For a real symmetric positive
definite matrix 
, it is
 |
It has unit row and column sums, so it is doubly stochastic iff 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 and mathematical development.
