Affine spaces of symmetric or alternating matrices with bounded rank

Pazzis, Clément de Seguins

doi:10.1016/j.laa.2016.04.026

Cited by 6 publications

(7 citation statements)

References 15 publications

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“…The existence of v with full tensor rank within the range of A ∈ M k ⊗ M k is essential for our algorithm to work. There are several possible properties to impose on a subspace of C k ⊗ C k in order to guarantee the existence of such v within this subspace [21][22][23][24][25][26][27]. The Edmonds-Rado property [1,2,28] is the most relevant to our problem.…”

Section: Introductionmentioning

confidence: 99%

Sinkhorn–Knopp theorem for PPT states

Cariello

2019

Lett Math Phys

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with tensor rank k, we provide an algorithm that checks whether the positive mapis equivalent to a doubly stochastic map. This procedure is based on the search for Perron eigenvectors of completely positive maps and unique solutions of, at most, k unconstrained quadratic minimization problems. As a corollary, we can check whether this state can be put in the filter normal form. This normal form is an important tool for studying quantum entanglement. An extension of this procedure to PPT states in M k ⊗ M m is also presented. √ k Id, D 1 = 1 √ m Id and tr(C i C j ) = tr(D i D j ) = 0, for every i = j [5, 6].

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Section: Introductionmentioning

confidence: 99%

Sinkhorn–Knopp theorem for PPT states

Cariello

2019

Lett Math Phys

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“…The following result was first obtained in [5] under the assumption that S is a linear subspace of A n (F) and |F| ≥ n + 1. The general case has recently been proved by de Seguins Pazzis [6]. Here it is obtained as a direct consequence of Theorem 1.2.…”

Section: Remarksmentioning

confidence: 70%

“…1. As noted above, de Seguins Pazzis [6] proved that Theorem 1.5 holds also over F 2 if S is an affine subspace of H n (F 2 ). It would be interesting to decide whether this case can also be handled using the combinatorial approach of the present paper.…”

Section: Discussionmentioning

confidence: 83%

“…The following result was first obtained in [5] under the assumption that S is a linear subspace of H n (F) and |F| ≥ n + 1. It has recently been proved by de Seguins Pazzis [6] for affine subspaces S of H n (F) with no restrictions on F. Here it is established for slightly more general affine spaces provided that F = F 2 .…”

Section: Remarksmentioning

confidence: 83%

“…Theorem 1.4 ( [5,6]). Let F be any field and let S be an affine subspace of A n (F) such that ρ(S) = k < n. Then dim S ≤ u a (n, k).…”

Section: Remarksmentioning

confidence: 99%

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Maximal rank in matrix spaces via graph matchings

Meshulam

2017

Linear Algebra and its Applications

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Let M n (F) be the space of n×n matrices over a field F. A matrix A = A(i, j) n i,j=1 ∈ M n (F) is weakly symmetric if A(i, j) = 0 iff A(j, i) = 0 holds for all i, j. A matrix is alternating if it is skew-symmetric with zero diagonal. Let W n (F) and A n (F) denote respectively the set of weakly symmetric matrices and the space of alternating matricesFor a translate S of a linear space B ⊂ W n (F) let G S be the graph with loops on the vertex set [n] with edge set E S = {q(B) : 0 = B ∈ B}. A subset M ⊂ E S is a matching if e ∩ e ′ = ∅ for all e = e ′ ∈ M . Let µ(G S ) = max e∈M |e| where M ranges over all matchings M ⊂ E S . Let ρ(S) denote the maximal rank of a matrix in S. It is shown that if S is a translate of a linear space contained in W n (F) and |F| ≥ 3 then ρ(S) ≥ µ(G S ). The restriction on F can be removed if S is an affine subspace of A n (F). Applications include simple proofs of upper bounds on the dimension of affine subspaces of symmetric and alternating matrices of bounded rank. 2000 MSC: 05C50; 47L05

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Large spaces of symmetric or alternating matrices with bounded rank

Pazzis

2016

Linear Algebra and its Applications

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Affine spaces of symmetric or alternating matrices with bounded rank

Cited by 6 publications

References 15 publications

Sinkhorn–Knopp theorem for PPT states

Sinkhorn–Knopp theorem for PPT states

Maximal rank in matrix spaces via graph matchings

Large spaces of symmetric or alternating matrices with bounded rank

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