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A counterexample to strict spectral convergence

Abstract

Schatten-p approximants need not converge to the strict spectral approximant. A complex linear subspace of three-by-three matrices has a strict residual with singular values (1,1,0), whereas every minimizing residual for sufficiently large p has real last diagonal entry less than -1/4.

Theorem 1.1 (Real powers of a square).

Proof. Machine-checked in Lean as D5/S3/Quantum/NormCompression/StrictSpectralPolyaRefutation.rpow_square_half (✓ std3). ∎

Source. Repository-derived.

Commentary.

For real a and p, the nonnegative square a² has power p/2 equal to the p-th power of |a|.

Definition 1.2 (Schatten-p approximation).

Formalization. D5/S3/Quantum/NormCompression/StrictSpectralPolyaRefutation.IsCpApproximant (✓ std3).

Citation. Krystyna Ziętak (2017). From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix. DOI: 10.4064/bc112-0-16. URL: https://doi.org/10.4064/bc112-0-16.

Commentary.

Page 308, verbatim: “Thus for 1 < p < ∞ the cₚ-approximant B = Bₚ ∈ ℳ to A satisfying (1.1) is unique.” Equation (1.1) minimizes the Schatten p-norm of A - X over X in the linear subspace. Here p is real, matrices have row and column carriers Fin m and Fin n, and schattenNorm is the finite singular-value norm in UpperBranchReduction. The limiting assertion below requires p > 1.

Definition 1.3 (Lexicographic spectral approximation).

Formalization. D5/S3/Quantum/NormCompression/StrictSpectralPolyaRefutation.IsStrictSpectralApproximant (✓ std3).

Citation. Krystyna Ziętak (2017). From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix. DOI: 10.4064/bc112-0-16. URL: https://doi.org/10.4064/bc112-0-16.

Commentary.

Page 309, verbatim: “A matrix B⁽ˢᵗ⁾ ∈ ℳ is a strict spectral approximant to a matrix A if the vector σ(R⁽ˢᵗ⁾) of the ordered singular values of the residual matrix R⁽ˢᵗ⁾ = A − B⁽ˢᵗ⁾ is minimal with respect to the lexicographic ordering in the set of vectors {σ ∈ ℝᵗ : σ = σ(A − X), X ∈ ℳ} (see [106]).” Fin m and Fin n index the entries. LinearMap.singularValues supplies decreasing ordered singular values, padded by zeros; toLex uses the natural order of ℕ.

Definition 1.4 (Ziętak’s Conjecture 1.1).

Formalization. D5/S3/Quantum/NormCompression/StrictSpectralPolyaRefutation.claim (✓ std3).

Citation. Krystyna Ziętak (2017). From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix. DOI: 10.4064/bc112-0-16. URL: https://doi.org/10.4064/bc112-0-16.

Commentary.

Page 309, Conjecture 1.1, verbatim: “Let ℳ be a linear subspace of ℂᵐˣⁿ and let Bₚ ∈ ℳ be the cₚ-approximant to A ∈ ℂᵐˣⁿ. As p tends to ∞, the matrix Rₚ = A − Bₚ converges to A − B⁽ˢᵗ⁾, where B⁽ˢᵗ⁾ is the strict spectral approximant to A.” The quantifiers range over all dimensions, complex linear subspaces, targets, strict approximants and functions B of the real exponent p. Minimization is required for every p > 1, and the residual converges in the matrix topology as p tends to infinity.

Theorem 1.5 (The universal convergence claim is false).

Proof. Machine-checked in Lean as D5/S3/Quantum/NormCompression/StrictSpectralPolyaRefutation.result (✓ std3). ∎

Resolves. Problems/zietak-2017-strict-spectral-polya-conjecture (refuted) by D5/S3/Quantum/NormCompression/StrictSpectralPolyaRefutation.result.

Source. Repository-derived.

Acknowledgement. Krystyna Ziętak (2017). From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix. DOI: 10.4064/bc112-0-16. URL: https://doi.org/10.4064/bc112-0-16.

Commentary.

Set A = I₃, H = [[1,0,1],[0,-2,0],[1,0,0]], E = Matrix.diagonal (Pi.single 2 1), and M = Submodule.span ℂ {H,E}. The residuals are [[1+s,0,s],[0,1-2s,0],[s,0,t]]. The two leading diagonal constraints force s = 0 on the spectral-optimal face, whose lexicographic minimum is diag(1,1,0), so E is strict. Schatten-p minimizers exist for every p > 1. Their real last residual entry is eventually below -1/4, contradicting convergence to the strict residual. Hermitian projection reduces the complex minimization to a real comparison. No uniqueness or exact residual limit is needed.

Grover–Gupta’s Theorem 3.7 for min(m,n) ≤ 2 remains valid. The three-dimensional example uses a repeated leading singular-value cluster and a coupling that does not commute with E. Eventual superiority against each fixed competitor does not give a uniform comparison against moving minimizers.

Priyanka Grover and Krishna Kumar Gupta, Properties of best approximations with respect to the Ky Fan p-k norm, and the strict spectral approximant of a matrix, LAA 743 (2026), DOI 10.1016/j.laa.2026.04.016; arXiv:2603.07498v2.

References