Bapat–Sunder's largest-eigenvalue conjecture for the permanent matrix

About 4 years old · traced to 2022

Let $\mathcal{H}_n$ be the set of $n\times n$ positive semidefinite Hermitian matrices. For $A=[a_{i,j}]\in\mathcal{H}_n$ , let $A(i,j)$ be the submatrix obtained by deleting row $i$ and column $j$ , and define $F_A=[f_{i,j}]$ by

f_{i,j}=a_{i,j}\operatorname{per}(A(i,j)).

Bapat–Sunder's conjecture. The largest eigenvalue of $F_A$ should satisfy

\lambda_{\max}(F_A)\leq \operatorname{per}(A).

Equivalently, $\operatorname{per}(A)$ should be the largest eigenvalue of $F_A$ . Drury disproved this conjecture, so it is refuted.

References

Primary source

Léo Pioge, Kamil K. Pietrasz, Benoit Seron, Leonardo Novo and Nicolas J. Cerf, “A logical implication between two conjectures on matrix permanents”, arXiv:2508.00111 (2025).

Additional references

2 papers in this index state this conjecture (2022–2025). The statement above is taken from the most recent of them; the others are arXiv:2202.01867.

matrix-eigenvalues matrix-permanents positive-semidefinite-matrices

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A counterexample by Drury refutes the conjecture, while the broader question of which matrices still satisfy the bound remains open.

The conjecture, attributed to Bapat and Sunder, asserts that the permanent gives the top eigenvalue of the associated matrix for every positive semidefinite Hermitian matrix. It had remained open for about three decades before Drury produced a counterexample.

Known results

The conjecture holds for $n\leq 3$ (reported in 2016).
Drury’s complex counterexample is a $7\times 7$ positive semidefinite correlation matrix of rank $2$ , with $\operatorname{per}(A)=45$ but $\lambda_{\max}(F_A)=525/8$ (reported in 2016).

August 2025 update

A 2025 paper states that Drury disproved the conjecture and distinguishes it from the stronger permanent-on-top conjecture: the latter implies this conjecture, but not conversely. It notes that identifying matrix classes where the bound remains valid is open.

Current status (as of September 2026): The conjecture is reported refuted by Drury’s counterexample; the characterization of matrix classes for which it remains valid is open.

Sources

No solutions have been posted yet.