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 $$ \min_{W\ \mathrm{nonsingular}}\|W\circ G\|_\mathrm{ui}\,\|W^{-1}\circ G^\mathrm{T}\|_\mathrm{ui} \quad\mbox{and}\quad \min_{P\ \mathrm{permutation}}\|P\circ G\|_\mathrm{ui}^2, $$ where $\Vert \cdot {\Vert }_{\mathrm{ui}}$ 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 $\Vert \cdot {\Vert }_{\mathrm{ui}}$ must be uniformly equivalent to the Frobenius norm $\Vert \cdot {\Vert }_{\mathrm{F}}$. We prove that the two quantities coincide when $\Vert \cdot {\Vert }_{\mathrm{ui}}=\Vert \cdot {\Vert }_{\mathrm{F}}$: for every complex matrix $G$, $$\underset{W\mathrm{nonsingular}}{\min }\Vert W\circ G{\Vert }_{\mathrm{F}}\Vert W^{-1}\circ G^{\mathrm{T}}{\Vert }_{\mathrm{F}}=\underset{P\mathrm{permutation}}{\min }\Vert P\circ G{\Vert }_{\mathrm{F}}^2.$$ The associated conjecture therefore holds with the optimal constant $c=1$.