On a Conjecture Related to Eigenvalue Perturbations

Abstract: In connection with eigenvalue perturbation bounds, Li [Linear Algebra Appl., 278 (1998), pp. 317-326] formulated three minimization problems involving Hadamard products. The third asks how to relate $$ \min_{W\ \mathrm{nonsingular}}|W\circ G|<em>\mathrm{ui}\,|W^{-1}\circ G^\mathrm{T}|</em>\mathrm{ui} \quad\mbox{and}\quad \min_{P\ \mathrm{permutation}}|P\circ G|<em>\mathrm{ui}^2, $$ where $|\cdot|</em>\mathrm{ui}$ denotes a unitarily invariant norm and $\circ$ denotes the Hadamard product, i.e., entrywise multiplication. The first quantity never exceeds the second, and a reverse inequality with a constant independent of the matrix size was conjectured. Romeo and Tilli [Linear Algebra Appl., 326 (2001), pp. 161-172] showed that if such a bound holds for every $G$, then $|\cdot|<em>\mathrm{ui}$ must be uniformly equivalent to the Frobenius norm $|\cdot|</em>\mathrm{F}$. We prove that the two quantities coincide when $|\cdot|<em>\mathrm{ui}=|\cdot|</em>\mathrm{F}$: for every complex matrix $G$, $$\min_{W\ \mathrm{nonsingular}} |W\circ G|<em>\mathrm{F}\,|W^{-1}\circ G^\mathrm{T}|</em>\mathrm{F} =\min_{P\ \mathrm{permutation}}|P\circ G|_\mathrm{F}^2.$$ The associated conjecture therefore holds with the optimal constant $c=1$.