Zhang–Li rank-one Hadamard product variational problem

Lei-Hong Zhang and Ren-Cang Li, On a Conjecture Related to Eigenvalue Perturbations, arXiv:2609.25552v1, §1 Problem 3 and §5:

Problem 3: Relate min_{W nonsingular} ‖W∘G‖_ui ‖W^{-1}∘G^T‖ui to min{P permutation} ‖P∘G‖_ui^2.

In fact, for rank-one G, Romeo and Tilli [23, Theorem 1.2] proved (1.1) with c1=1 and the unitary restriction (2.3) with c2=1. They explicitly left the corresponding nonsingular version of Problem 3 unresolved; see [23, Remark 3.1].

For every natural $n$, every complex $n\times n$ matrix $G$ with $\operatorname{rank}(G)\le1$, and every unitarily invariant norm $N$, the precise conclusion is

$$ \inf_{W\in\mathrm{GL}n(\mathbb C)} N(W\circ G)N(W^{-1}\circ G^{\mathsf T}) =\min{P\text{ permutation}}N(P\circ G)^2. $$

A permutation minimizing $N(P\circ G)$ attains the infimum. The transpose of $G$ is plain transpose. The zero matrix, zero weights, and $n=0$ are included.

The question compares arbitrary changes of basis with discrete permutation matchings in a norm expression associated with eigenvalue perturbations. Equality with constant one would show that nonsingular matrices cannot improve the rank-one permutation minimum, extending the cited unitary case. Zhang–Li Theorem 1.1 already proves Conjecture 1.1 with $c=1$ for the Frobenius norm for every $G$, without a rank restriction. The rank-one question here concerns every unitarily invariant norm.

The cited Romeo–Tilli unitary result does not cover arbitrary nonsingular $W$. Singular values of such a $W$ can lie on either side of one. A common lower bound must therefore connect the two factors involving $W$ and $W^{-1}$ without imposing a unitary restriction. The classical background is the lower Gelfand–Naimark inequality and symmetric-gauge norm comparison; the required link is an expanding middle factor and its balanced average. Zhang–Li Theorem 1.1 settles the Frobenius-norm case for every $G$ with $c=1$; the missing case is the rank-one assertion for general unitarily invariant norms. The source identifies this rank-one gap in §5. It is a first-tier recent published question in the sense of the repository's problem classification; the absence of later solutions is not independently established here.

E1: expanding average. For invertible $W$ and $t>0$, set

$$ T_t=\frac{tW+t^{-1}W^{-}}2,\qquad W^{-}=(W^{-1})^*. $$

Writing $S_t=tW-t^{-1}W^{-*}$ gives $T_t^*T_t-I=S_t^*S_t/4\succeq0$. This identity supplies expansion for every positive scale. R. Bhatia, F. Kittaneh, R.-C. Li, “Some inequalities for commutators and an application to spectral variation. II”, Linear and Multilinear Algebra 43 (1997), no. 1–3, 207–219, https://doi.org/10.1080/03081089708818526, §4, provides the related diagonal matrix-and-inverse averaging precedent.

E2: expanding diagonal minimum. For nonnegative real diagonal weights $a,b$, a single permutation $P_0$ satisfies

$$ N(\operatorname{diag}(a)P_0\operatorname{diag}(b)) \le N(\operatorname{diag}(a)T\operatorname{diag}(b)) \quad(T^*T\succeq I). $$

For positive weights, put $C=\operatorname{diag}(a)T\operatorname{diag}(b)$ and $D=\operatorname{diag}(1/b)$. The lower Gelfand–Naimark inequality on arbitrary index subsets bounds products of $s_i(CD)/s_i(D)$ by initial products of $s(C)$. Expansion gives $s_i(CD)=s_i(\operatorname{diag}(a)T)\ge a_i^\downarrow$ and $s_i(D)=1/b_i^\uparrow$. Consequently the opposite matching $x_i=a_i^\downarrow b_i^\uparrow$ satisfies, for every index subset $I$,

$$ \prod_{i\in I}x_i\le\prod_{j=1}^{|I|}s_j(C). $$

Sort $x$ decreasingly. Its initial products now satisfy weak log-majorization, which implies weak majorization through the initial sum inequalities. The symmetric gauge representation $N(X)=\gamma(s(X))$ and Fan dominance give the norm comparison. A permutation realizing this opposite matching gives $P_0$. Replacing both weights by $a+\varepsilon$ and $b+\varepsilon$ preserves their sorting permutations. Continuity as $\varepsilon\downarrow0$ extends the same permutation to zero weights.

E3: optimization in t. Put $A=\operatorname{diag}(a)$, $B=\operatorname{diag}(b)$ and $m=N(AP_0B)$. E1 and E2, the triangle inequality, and homogeneity give, for every $t>0$,

$$ 2m\le tN(AWB)+t^{-1}N(AW^{-*}B). $$

For nonnegative $m,f,g$, these bounds imply $m^2\le fg$ with $f=N(AWB)$ and $g=N(AW^{-*}B)$. If $m,f>0$, take $t=m/f$; if $f=0<m$, take $t=(g+1)/m$ to obtain a contradiction. Thus the product bound includes vanishing factors.

E4: phases and attainment. Factor $G=xy^{\mathsf T}$ and set $a_i=|x_i|$, $b_i=|y_i|$. Removing diagonal unitary phases gives, simultaneously for all $W$,

$$ N(W\circ G)=N(AWB),\qquad N(W^{-1}\circ G^{\mathsf T})=N(AW^{-*}B). $$

Transpose and adjoint invariance give the second identity; choose phase one at zero entries. Every permutation matrix is expanding, so E2 also proves that $P_0$ minimizes the permutation norm. Finally, $P_0^{-1}\circ G^{\mathsf T}=(P_0\circ G)^{\mathsf T}$, giving equality of the two factors at $P_0$ and hence attainment.

A violation would consist of a dimension, rank-at-most-one complex $G$, a unitarily invariant norm, and an invertible $W$ for which the product is strictly below every squared permutation norm. A failure of attainment at a minimizing permutation would also contradict the statement. Correctness of this quantified assertion is separate from bibliographic priority.

The settling declaration is result : claim in D5/S3/Quantum/Schatten/ZhangLiRankOneProblemThree.lean. Its claim provides one permutation, the comparison with every permutation, the product lower bound for every invertible matrix, and equality at that permutation.

| Mathematical step | Declaration | | --- | --- | | Balanced expansion | inverseAdjointAverage_expanding | | Arbitrary-subset ratio bound | lower_gelfand_naimark_ratio | | Singular-value growth under expansion | expanding_right_singularValuesSorted | | Subset products to diagonal gauges | UINorm.N_diagonal_subsetProducts_le | | Common expanding minimum | UINorm.exists_permutation_expanding_lower_bound | | Balanced product bound | UINorm.sq_le_product_of_expanding_lower_bound | | Rank-one phase reduction | UINorm.rank_one_hadamard_reduction |

These declarations express E1–E4. This dossier does not assert an additional formalized transfer theorem or a solution in higher rank.

The rank-at-most-one statement is PROVED by result : claim, including the exact constant one and attainment. The decisive mechanism is a common permutation lower bound on expanding matrices, coupled to an inverse-adjoint average which is expanding at every positive scale. Allowing zero weights requires continuity, and adds the zero matrix without weakening the result.

The following transfer principle is proved at paper level, not formalized. Fix a real matrix $H$, a unitarily invariant norm $N$, and $c\ge0$, and let $m_H=\min_P N(P\circ H)$. If, for every invertible complex $Z$, the one-sided estimate

$$ m_H\le c|Z^{-1}|_{\mathrm{op}}N(Z\circ H) $$

holds, then every invertible complex $W$ satisfies

$$ m_H^2\le c^2 N(W\circ H)N(W^{-1}\circ H^{\mathsf T}). $$

To prove it, apply the hypothesis to the expanding matrix $T_t$ of E1. Expansion gives $|T_t^{-1}|_{\mathrm{op}}\le1$. Since $H$ is real, $W^{-}\circ H=(W^{-1}\circ H^{\mathsf T})^$, so adjoint invariance and the triangle inequality give $2m_H\le c(tN(W\circ H)+t^{-1}N(W^{-1}\circ H^{\mathsf T}))$ for every $t>0$. The scalar argument of E3, with factors multiplied by $c$, proves the assertion, including $c=0$ and zero factors. This consequence does not supply the one-sided estimate for any additional $H$.

The rank-at-least-two question in Zhang–Li §5 remains OPEN for general unitarily invariant norms. For the Frobenius norm it is solved by Zhang–Li Theorem 1.1, which proves Conjecture 1.1 with $c=1$ for every $G$. The common diagonal reduction here uses rank at most one; the result does not extend that reduction to higher rank. No other conjecture or downstream result is claimed as settled.

The exact remark number for the Bhatia–Kittaneh–Li §4 averaging precedent is ASSUMED-UNVERIFIED: “Remark 2” was reported through a secondary reading and has not been independently inspected.

Romeo–Tilli Remark 3.1 wording has been read only through Zhang–Li's citation. The original remark has not been independently inspected. Citation indexes and subsequent literature have not been exhaustively checked, and the absence of a later solution is not established. The source quotations and metadata are supplied material; no online verification is included. The transfer principle in Triage is a paper-level consequence and has no corresponding formalized declaration in this delivery.