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Inverse relative gain array conjecture

For every integer n≥1n\ge 1 and every real symmetric positive definite matrix G∈Rn×nG\in\mathbb{R}^{n\times n}, the matrix (G∘G−1)−1\left(G\circ G^{-1}\right)^{-1} is entrywise nonnegative, where ∘\circ denotes the Hadamard product; equivalently, (G∘G−1)ij−1≥0\left(G\circ G^{-1}\right)^{-1}_{ij}\ge 0 for all 1≤i,j≤n1\le i,j\le n.

References

Primary source

arXiv

Additional references

inequalitiesmatrix-analysis
Posted · edited

Progress summary

Refreshed
Claimed solved

A recent preprint claims the conjecture is true through six dimensions but gives a rational counterexample in seven, so the unrestricted conjecture is false.

The conjecture concerns entrywise nonnegativity of the inverse relative gain array for real symmetric positive definite matrices. The reported result identifies six as the largest universally valid order and rules out the unrestricted claim from order seven onward.

September 28, 2026 sharp-dimension result

Jie Wang's preprint claims that the conjecture holds for every real symmetric positive definite matrix of order at most six, and provides a rational order-seven counterexample. This settles the sharp range if the proof and counterexample are correct, but the claim is unverified in the retrieved material.

Current status (as of September 2026): The conjecture is claimed, but not independently verified here, to hold through order six and fail in order seven; the unrestricted higher-order statement is therefore claimed false.

Sources

Solutions 0

No solutions have been posted yet.