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A \(3\times 3\) counterexample to Lin and Wimmer's rank-minimization conjecture associated with Roth's similarity theorem
Shuo Shi, Juan Zhang, Yun Zhang
TL;DR
The note examines Conjecture 1.2 and shows it is false without additional spectral hypotheses. The authors give an explicit 3 × 3 counterexample and prove that, over C, if min{m, n} < 3, identity (3) holds for every C. The rank minima are 2 and 1 in the 3 × 3 example, while over C no counterexample occurs when min{m, n} < 3.
Problem
The note examines Conjecture 1.2 and shows it is false without additional spectral hypotheses.
Source: §1 · p. 2Method
Results
Takeaways & Limitations
The construction is dimensionally minimal over C, and the authors identify arbitrary invertible P as allowing a lower rank than block upper triangular intertwiners can attain.
Source: §5 · p. 6Unless otherwise stated, the paper considers only cases where A and B share common eigenvalues; without common eigenvalues, both rank quantities are zero.
Source: §2 · p. 3
Abstract
from arXiv · showhide
We give a \(3\times 3\) counterexample, valid over every field, to a rank-minimization conjecture of Lin and Wimmer (Bull. Aust. Math. Soc., 84 (3) (2011), 441--443) related to Roth's similarity theorem for the Sylvester matrix equation. We also prove that, over the complex field, no counterexample can occur when one of the two matrix sizes is less than \(3\). Hence the example is dimensionally minimal over the complex field.
1 Introduction
Following work on Roth’s similarity theorem and generalized Sylvester equations, Lin and Wimmer proposed a rank-minimization analogue. This note disproves that conjecture with a 3 × 3 example and establishes minimality over the complex field.
1 Introduction
- Roth’s similarity theorem characterizes solvability of the Sylvester matrix equation through similarity of associated matrices.Over C, the equation has a unique solution for every C if and only if A and B have no common eigenvalue.Source: §1 · p. 1
- Lin and Wimmer showed that Roth’s equivalence theorem is a special case of a rank-minimization theorem, then proposed a related conjecture for Roth’s similarity theorem.Ito and Wimmer had extended this direction to generalized Sylvester equations over B´ezout domains.Source: §1 · p. 2
- Ferrante and Wimmer proved the conjectured equality under certain spectral hypotheses.The note’s counterexample concerns the case without those additional hypotheses.Source: §1 · p. 2
- Conjecture 1.2 is false without additional spectral hypotheses: the note gives an explicit 3 × 3 counterexample, and over C proves this size minimal because identity (3) holds whenever min{m, n} < 3.Source: §1 · p. 2
2 Background and quoted results
The background section relates the two rank quantities, recalls when equality is known to hold, and specifies the spectral setting considered in the paper.
2 Background and quoted results
- Conjecture 1.2 asserts α(A, B, C) = γ(A, B, C) for all triples, with γ(A, B, C) defined as the minimum rank of PMC − M0P over invertible P.The elementary comparison between these quantities is identified as the starting point of the problem.Source: §2 · p. 2
- An eigenvalue is semisimple when its geometric and algebraic multiplicities agree, and nonderogatory when its geometric multiplicity is 1.Equivalently, semisimplicity means all corresponding Jordan blocks have size 1, while nonderogatory means there is exactly one such block.Source: §2 · p. 2
- Ferrante and Wimmer [1] proved equality when each common eigenvalue is semisimple for both A and B.Their theorem also gives a separate sufficient condition involving nonderogatory eigenvalues.Source: §2 · pp. 2–3
- The paper considers triples where A and B share eigenvalues; without common eigenvalues, α(A, B, C) = γ(A, B, C) = 0.The equality in the zero case follows from Roth’s similarity theorem and Proposition 2.1.Source: §2 · pp. 2–3
3 The counterexample
The authors construct matrices over an arbitrary field for which the two rank minima differ. Their comparison gives a 3 × 3 counterexample to the conjecture.
3 The counterexample
- The example’s only eigenvalue is 0, with algebraic multiplicity 3 and geometric multiplicity 2, so it is neither semisimple nor nonderogatory.It therefore lies outside both alternatives in Theorem 2.3.Source: §3 · p. 3
- To bound the left-hand minimum, the authors show that the columns v1, v2, v3 cannot all be scalar multiples of v3.The third component of v3 is nonzero, forcing v1 to be zero if it is a multiple of v3; the second-component condition then gives a contradiction.Source: §3 · pp. 3–4
- The matrix used on the right-hand side has rank 1, and the authors show the minimum cannot be 0.Together these establish the right-hand minimum used in the strict inequality.Source: §3 · p. 4
- min X∈K^3×3 rank(AX − XB − C) = 2, while min P∈GL(6,K) rank(PMC − M0P) = 1, disproving Conjecture 1.2.The construction is over an arbitrary field.Source: §3 · pp. 3–4
4 Dimensional minimality over the complex field
Over C, identity (3) holds whenever min{m, n} < 3, establishing that the 3 × 3 counterexample is dimensionally minimal. The proof treats scalar cases directly and reduces non-scalar 2 × 2 cases to nonderogatory eigenvalues.
4 Dimensional minimality over the complex field
- Over C, identity (3) holds whenever min{m, n} < 3.Thus the 3 × 3 counterexample is dimensionally minimal over the complex field.Source: §4 · p. 5
- For m = 1, the scalar case applies; for non-scalar A of size 2, every eigenvalue is nonderogatory, so Theorem 2.3 gives identity (3).A non-scalar 2 × 2 complex matrix has every eigenvalue nonderogatory.Source: §4 · p. 5
- The case n < 3 follows by a similar argument.Source: §4 · p. 5
- If either A or B is scalar, Lemma 4.1 gives equality, even where Theorem 2.3’s spectral hypotheses may fail.Theorem 2.3’s nonderogatory or semisimple-both-sides conditions are sufficient, not necessary.Source: §4 · p. 5
5 Conclusion
The paper gives a 3 × 3 counterexample over an arbitrary field to Lin and Wimmer’s rank-minimization conjecture. It notes that unrestricted invertible intertwiners can attain lower rank than block upper triangular ones, and that the example is dimensionally minimal over C.
5 Conclusion
- A 3 × 3 counterexample over an arbitrary field disproves Lin and Wimmer’s rank-minimization conjecture for Roth’s similarity theorem.Source: §5 · p. 6
- Allowing arbitrary invertible P can reduce rank below what block upper triangular intertwiners attain.Source: §5 · p. 6
- Over C, the phenomenon cannot occur when either matrix size is below 3; characterizing when γ(A, B, C) < α(A, B, C) remains open.Source: §5 · p. 6