[Paper Review] Remarks on BMV conjecture

Shmuel Friedland|ArXiv.org|Apr 24, 2008

Matrix Theory and Algorithms3 references3 citations

TL;DR

This paper establishes an asymptotic positivity result for coefficients of the $ t^k $-term in the trace polynomial $ \mathrm{Tr}(A + tB)^m $, where $ A $ and $ B $ are positive semidefinite Hermitian matrices. It proves that for fixed $ k $ and $ \mathrm{Tr}(AB) > 0 $, the coefficient of $ t^k $ is positive when $ m $ exceeds a threshold $ N(A,B,k) $, and shows that the leading-order asymptotic behavior is governed by the trace of the $ k $-th power of the principal submatrix of $ B $ corresponding to the largest eigenvalues of $ A $.

ABSTRACT

We show that for fixed $A,B$, hermitian nonnegative definite matrices, and fixed $k$ the coefficients of the $t^k$ in the polynomial $ r (A+tB)^m$ is positive if $ r AB >0$ and $m>N(A,B,k)$.

Motivation & Objective

• To establish an asymptotic positivity result for individual coefficients in the trace polynomial $ \mathrm{Tr}(A + tB)^m $, under the condition $ \mathrm{Tr}(AB) > 0 $.
• To determine the threshold $ m > N(A,B,k) $ beyond which the coefficient of $ t^k $ becomes strictly positive.
• To analyze the leading-order asymptotic behavior of $ \mathrm{Tr}(S_{m,k}(A,B)) $ as $ m \to \infty $, identifying the dominant contribution from the largest eigenvalue block of $ A $ and corresponding submatrix of $ B $.
• To verify the nonnegativity of the $ t^3 $-coefficient in $ \mathrm{Tr}(A + tB)^m $ for all $ m $ in the case $ n = 3 $, under specific constraints on $ A $ and $ B $.

Proposed method

• Utilizes recursive matrix trace identities via the recurrence $ S_{m+1,k} = A S_{m,k} + B S_{m+1,k-1} $, with $ S_{p,q} = 0 $ for $ \min(p,q) < 0 $, to express $ \mathrm{Tr}(S_{m,k}(A,B)) $ as a coefficient in a generating function.
• Applies the Neumann series expansion to the resolvent of a block Toeplitz matrix $ T_k(A,B) $, linking the generating function of $ \mathrm{Tr}(S_{m,k}(A,B)) $ to the trace of $ (I - tA)^{-1} (B(I - tA)^{-1})^k $.
• Analyzes the rational generating function $ \mathrm{Tr}(I - tA)^{-1} (B(I - tA)^{-1})^k $, identifying poles at $ z = 1/a_i $, and computes the residue at the dominant pole $ z = 1/a_p $, where $ a_p $ is the largest positive eigenvalue of $ A $.
• Establishes that the residue at $ z = 1/a_p $ is $ \mathrm{Tr}(C^k) $, where $ C $ is the principal submatrix of $ B $ corresponding to the indices where $ a_i = a_p $, and uses this to derive the asymptotic formula.
• Employs the cyclic invariance of the trace and the nonnegativity of $ \mathrm{Tr}(CD) $ for $ C,D \succeq 0 $ to analyze low-degree terms and verify nonnegativity in the $ n = 3, k = 3 $ case.
• Reduces the $ n = 3, k = 3 $ case to verifying nonnegativity of Taylor series coefficients of a rational function in $ t $, under the constraints $ \det B = 0 $, $ B \succeq 0 $, and $ A = \mathrm{diag}(1,a,0) $.

Experimental results

Research questions

• RQ1 For fixed $ k $, $ A $, and $ B $, does the coefficient of $ t^k $ in $ \mathrm{Tr}(A + tB)^m $ become positive for sufficiently large $ m $, provided $ \mathrm{Tr}(AB) > 0 $?
• RQ2 What is the asymptotic behavior of $ \mathrm{Tr}(S_{m,k}(A,B)) $ as $ m \to \infty $, and which submatrix of $ B $ governs the leading-order term?
• RQ3 Can the nonnegativity of the $ t^3 $-coefficient in $ \mathrm{Tr}(A + tB)^m $ be established for all $ m $ and all $ 3 \times 3 $ positive semidefinite $ A, B $?
• RQ4 How does the structure of $ A $ and $ B $, particularly the location of their maximal eigenvalues, affect the asymptotic positivity of the $ t^k $-coefficient?
• RQ5 What role does the vanishing of $ \mathrm{Tr}(AB) $ play in the vanishing of $ \mathrm{Tr}(S_{m,k}(A,B)) $?

Key findings

• For fixed $ k $, $ A $, and $ B $ with $ \mathrm{Tr}(AB) > 0 $, the coefficient of $ t^k $ in $ \mathrm{Tr}(A + tB)^m $ is positive for all $ m > N(A,B,k) $, establishing an asymptotic positivity result.
• The leading-order asymptotic behavior of $ \mathrm{Tr}(S_{m,k}(A,B)) $ as $ m \to \infty $ is $ \mathrm{Tr}(C^k) \cdot a_p^m \binom{m+k}{k} $, where $ C $ is the principal submatrix of $ B $ corresponding to the indices where $ a_i = a_p $, the largest positive eigenvalue of $ A $.
• The limit $ \lim_{m \to \infty} \frac{\mathrm{Tr}(S_{m,k}(A,B))}{a_p^m \binom{m+k}{k}} = \mathrm{Tr}(C^k) $ holds, and since $ C \succeq 0 $, it follows that $ \mathrm{Tr}(C^k) \geq b_{pp}^k > 0 $, ensuring positivity.
• When $ \mathrm{Tr}(AB) = 0 $, it follows that $ AB = 0 $, and hence $ \mathrm{Tr}(S_{m,k}(A,B)) = 0 $ for all $ m,k \geq 1 $, confirming the degenerate case.
• For $ n = 3 $, the coefficient of $ t^3 $ in $ \mathrm{Tr}(A + tB)^m $ is shown to be nonnegative for all $ m $, under the constraints $ A = \mathrm{diag}(1,a,0) $, $ B \succeq 0 $, $ \det B = 0 $, and $ u,v,w > 0 $, by analyzing the rational generating function and verifying nonnegativity of all Taylor coefficients.
• The proof relies on explicit computation of the diagonal entries of $ (B(I - tA)^{-1})^3 $, and shows that all terms in the resulting series for $ \mathrm{Tr}(I - tA)^{-1} (B(I - tA)^{-1})^3 $ have nonnegative coefficients under the given constraints.