Linear Algebra and Matrices

We present an elementary, self-contained proof of Grothendieck's inequality that unifies the real and complex cases and yields both the Krivine and Haagerup bounds, the current bestknown explicit bounds for the real and complex Grothendieck constants respectively. This article is intended to be pedagogical, combining and streamlining known ideas of Lindenstrauss-Pe lczyński, Krivine, and Haagerup into a proof that need only univariate calculus, basic complex variables, and a modicum of linear algebra as prerequisites.2010 Mathematics Subject Classification. 47A07, 46B28, 46B85, 81P40, 81P45, 03D15, 97K30, 47N10, 90C27.

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“…Proof. We will start with (8). Suppose there exists M ∈ F m×n such that the left-hand side of ( 8) exceeds the right-hand side.…”

Section: Gaussian Integral Of Sign Functionmentioning

confidence: 99%

“…is an orthonormal basis of (F n ) ⊗k . For more information about hypermatrices see [15] and for a more formal definition of tensor products see [8].…”

Section: A Unified Proof Of Grothendieck's Inequalitymentioning

confidence: 99%

An elementary and unified proof of Grothendieck’s inequality

2019

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“…where the norm on X, Y, M is the matrix Frobenius (i.e., Hilbert-Schmidt) norm. In this case, (18) µ l,m,n σ = 1, for all l, m, n ∈ N, since, by Cauchy-Schwartz and the submultiplicativity of the Frobenius norm,…”

Section: Grothendieck-hölder Inequalitymentioning

confidence: 99%

“…and equality is attained by choosing M, X, Y with 1 in the (1, 1)th entry and 0 everywhere else. We will use (18) to obtain lower bounds on µ l,m,n p,q,r below. ( 14) and (19) will collectively be referred to as the Grothendieck-Hölder inequality.…”

Section: Grothendieck-hölder Inequalitymentioning

confidence: 99%

Grothendieck constant is norm of Strassen matrix multiplication tensor

Zhang¹,

Lim²

2017

Preprint

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We show that two important quantities from two disparate areas of complexity theory -Strassen's exponent of matrix multiplication ω and Grothendieck's constant KG -are intimately related. They are different measures of size for the same underlying object -the matrix multiplication tensor, i.e., the 3-tensor or bilinear operator µ l,m,n : F l×m × F m×n → F l×n , (A, B) → AB defined by matrix-matrix product over F = R or C. It is well-known that Strassen's exponent of matrix multiplication is the greatest lower bound on (the log of) a tensor rank of µ l,m,n . We will show that Grothendieck's constant is the least upper bound on a tensor norm of µ l,m,n , taken over all l, m, n ∈ N. Aside from relating the two celebrated quantities, this insight allows us to rewrite Grothendieck's inequality as a norm inequalityKG.We prove that Grothendieck's inequality is unique: If we generalize the (1, 2, ∞)-norm to arbitrary p, q, r ∈ [1, ∞],then (p, q, r) = (1, 2, ∞) is, up to cyclic permutations, the only choice for which µ l,m,n p,q,r is uniformly bounded by a constant independent of l, m, n.

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“…The computation of the rank of matrix is usually obtained by applying the Gaussian elimination process: Namely, it is the number of non-zero rows in the row echelon form obtained from the Gaussian elimination process. See [11] for classical results on matrix rank.…”

Section: Introductionmentioning

confidence: 99%