Li’s Hadamard-product eigenvalue-perturbation conjecture

For every unitarily invariant matrix norm $\|\cdot\|_{\mathrm{ui}}$ , does there exist a constant $c>0$ , independent of $n$ and of $G\in\mathbb{C}^{n\times n}$ , such that

\min_{P\in\mathcal{P}_n}\|P\circ G\|_{\mathrm{ui}}^2\leq c\,\min_{W\in\mathrm{GL}_n(\mathbb{C})}\|W\circ G\|_{\mathrm{ui}}\,\|W^{-1}\circ G^{\mathsf{T}}\|_{\mathrm{ui}}?

Here $\mathcal{P}_n$ is the set of $n\times n$ permutation matrices and $\circ$ denotes the Hadamard product. The reverse inequality is known in the opposite direction without a constant: the second minimum is at most the first. In the Frobenius-norm case, equality holds for every $G$ , so the optimal constant is $c=1$ .

References

Primary source

arXiv

Additional references

On a Conjecture Related to Eigenvalue Perturbations — arXiv — Lei-Hong Zhang, Ren-Cang Li

linear-algebra matrix-analysis

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed progress

A new paper settles the conjecture for the Frobenius norm, but the broader question covering all unitarily invariant norms remains open.

Li’s conjecture asks whether two eigenvalue-perturbation minimization quantities coincide for every complex matrix and every unitarily invariant norm. A recent paper by Lei-Hong Zhang and Ren-Cang Li establishes equality in the Frobenius-norm case and determines the optimal constant there.

Known results

The proposed inequality holds for every Q-norm, including the Schatten $p$ -norms for $2\le p\le\infty$ .
It holds for every unitarily invariant norm when $\operatorname{rank}(E)\le 1$ .
A $2\times2$ example attains equality, showing sharpness of the bound.

Recent Frobenius-norm result

Zhang and Li claim the two minimization quantities coincide for every complex matrix under the Frobenius norm, with the optimal constant identified. Their result does not settle the conjecture for arbitrary unitarily invariant norms.

Current status (as of September 2026): The Frobenius-norm case and several restricted norm classes are claimed settled, while the universal unitarily invariant norm case remains open.

Sources

No solutions have been posted yet.