Metadata
bibkey: lin2016psimap
authors: Minghua Lin
year: 2016
title: "New properties for certain positive semidefinite matrices"
doi: 10.48550/arXiv.1609.08231
url: https://arxiv.org/abs/1609.08231v1
claim: "Example 3.6 asks whether the map Ψ(T)=2(tr T)I−T gives the a-property and g-property for every positive semidefinite two-by-two block matrix."
strata_touched:
- D5/S3/Quantum/Matrix/LinPsiMap/ReductionBlock
- D5/S3/Quantum/Matrix/LinPsiMap/GramDiscrepancy
- D5/S3/Quantum/Matrix/LinPsiMap/Result
license: citation-only
triage: anchor
The reduction map and Lin’s two spectral properties
Locator
DOI: https://doi.org/10.48550/arXiv.1609.08231
The fixed source is Minghua Lin, New properties for certain positive semidefinite matrices, arXiv:1609.08231v1, §3, Example 3.6, with the properties defined in Definition 1.1(iii)–(iv). The journal version is Linear Algebra and its Applications 520 (2017), 32–43. The concluding §5 also identifies Example 3.6 as challenging.
The setup states:
we assume that [A X; X∗ B], where A, X, B are n × n, is positive semidefinite.
Definition 1.1(iii) states:
M is said to have a-property if 2sj (M12 ) ≤ λj (M11 + M22 ), j = 1, . . . , n.
Definition 1.1(iv) states:
M is said to have g-property if sj (M12 ) ≤ λj (M11 ♯ M22 ), j = 1, . . . , n.
Example 3.6 states:
If we consider the map Ψ : X ↦ 2(trX)In − X, then using the approach in [14] we can show that the matrix [Ψ(A) Ψ(X); Ψ(X∗) Ψ(B)] is PPT. Though there are strong numerical evidence suggesting that this block matrix also has g-property, yet we have not even been able to show that it has a-property.
The concluding §5 states:
Besides the challenging problems described in Example 3.5 and Example 3.6, …
Mathematical scope
For every natural dimension n, complex n-by-n matrices A,X,B with positive semidefinite block matrix [A X; X* B], and every decreasing spectral index j, the two requested conclusions are
The geometric mean is the same total matrix definition used for trace augmentation. It has the standard inverse-square-root formula on positive definite inputs and vanishes when either input vanishes. Nonzero positive diagonal blocks have strictly positive trace, and their images under Ψ are positive definite. No commutation or rank assumption is imposed.
Literature-check scope and result
The supplied literature reading covers the paper title and arXiv number with Example 3.6, g-property, a-property and proof; the formulas “2(trX)I − X” and “2tr(X)I − X” with singular-value inequalities; and the reduction map with the matrix geometric mean and block matrices.
Yang–Lu–Chen, Filomat 34 (2020), treats the analogous map Ψ₁(T)=(2n+1)(tr T)I−T, whose coefficient is larger than the coefficient two here. Alakhrass, arXiv:2305.10461 (2023), obtains for PPT blocks an index-halved geometric-mean bound: its right-hand index is floor((j+1)/2), so it does not supply the matching-index g-property. Rani–Kapil–Singh, Advances in Operator Theory 8 (2023), was also included in the supplied reading. None of these readings supplies a settlement of Example 3.6. Citation indexes were not exhaustively checked.
Attribution boundary
The quotations, bibliographic metadata and concrete literature reading are supplied source information, ASSUMED-UNVERIFIED against original papers under the offline constraint. They do not certify literature-wide priority. The reduction-block positivity identity, the general Gram-discrepancy estimate and their application to the reflected spectrum are the supplied mathematical route; no proof of Example 3.6 is attributed to the named literature. This citation-only note incorporates no source software.