The Schur Product Theorem in the Block Case

Choudhury, D.

doi:10.2307/2047941

Cited by 3 publications

(11 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(i) This is a well-known result and one may find it in books like [8] and [40] where H jk 's are normal n × n matrices for 1 ≤ j, k ≤ m. Theorem 5 [16] says that if the m 2 matrices {H jk : 1 ≤ j, k ≤ m} are a commuting family and H is positive semi-definite then H α = [H α jk ] is positive semi-definite for all α = 1, 2, . .…”

Section: C Positivity Of Block Matrices and Schur Or Hadamard Prodmentioning

confidence: 91%

“…Power Symmetric Matrices and Construction of PPT States. This section has been motivated by the work on "Power symmetric matrics" by Sinkhorn [81] and its generalization by Bapat, Jain and Prasad [5] on one hand and preservation of positivity of a block matrix under taking powers of blocks, the so-called Schur or Hadamard product by Choudhury [16] and Guillot, Khare and Rajaratnam [34] on the other. The purpose is to illustrate the interesting interplay rather than the utmost generality.…”

Section: Now the Matrixmentioning

confidence: 99%

“…Jain and K. Manjunatha Prasad [5]. Techniques developed by D. Choudhury [16] and D. Guillot, A. Khare and B. Rajaratnam [34] help to some extent. The fourth section is devoted to the concept of unitary equivalence of a matrix to its transpose and its general form obtained by S.R.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Role Of Partial Transpose And Generalized Choi Maps In Quantum Dynamical Semigroups Involving Separable And Entangled States

Singh

2015

ELA

View full text Add to dashboard Cite

Abstract. Power symmetric matrices defined and studied by R. Sinkhorn (1981) and their generalization by R.B. Bapat, S.K. Jain and K. Manjunatha Prasad (1999) have been utilized to give positive block matrices with trace one possessing positive partial transpose, the so-called PPT states. Another method to construct such PPT states is given, it uses the form of a matrix unitarily equivalent to its transpose obtained by S.R. Garcia and J.E. Tener (2012). Evolvement or suppression of separability or entanglement of various levels for a quantum dynamical semigroup of completely positive maps has been studied using Choi-Jamiolkowsky matrix of such maps and the famous Horodecki's criteria (1996). A Trichotomy Theorem has been proved, and examples have been given that depend mainly on generalized Choi maps and clearly distinguish the levels of entanglement breaking.

show abstract

Section: C Positivity Of Block Matrices and Schur Or Hadamard Prodmentioning

confidence: 91%

Section: Now the Matrixmentioning

confidence: 99%

See 1 more Smart Citation

Role Of Partial Transpose And Generalized Choi Maps In Quantum Dynamical Semigroups Involving Separable And Entangled States

Singh

2015

ELA

View full text Add to dashboard Cite

show abstract

“…Numerical experiment suggests that in general H fails to be positive semidefinite for any finite p > 2. We borrow the following example from [7] to show that the result is also not true in general for p = ∞.…”

Section: Multiplying Both Sides By Detmentioning

confidence: 99%

Positive semidefinite 3 x 3 block matrices

Lin

Driessche²

2014

ELA

View full text Add to dashboard Cite

show abstract

“…A natural generalization of this problem consists of studying powers preserving positivity when applied to block matrices (see e.g. [4,14,28]). More precisely, let H := (H st ) n s,t=1 be an mn×mn Hermitian positive semidefinite matrix, where each block H st is an m × m Hermitian positive semidefinite matrix.…”

Section: Introductionmentioning

confidence: 99%

On fractional Hadamard powers of positive block matrices

Guillot¹,

Khare²,

Rajaratnam³

2014

Preprint

View full text Add to dashboard Cite

Entrywise powers of matrices have been well-studied in the literature, and have recently received renewed attention due to their application in the regularization of highdimensional correlation matrices. In this paper, we study powers of positive semidefinite block matrices (Hst) n s,t=1 where each block Hst is a complex m × m matrix. We first characterize the powers α ∈ R such that the blockwise power map (Hst) → (H α st ) preserves Loewner positivity. The characterization is obtained by exploiting connections with the theory of matrix monotone functions which was developed by C. Loewner. Second, we revisit previous work by D. Choudhury [Proc. Amer. Math. Soc. 108] who had provided a lower bound on α for preserving positivity when the blocks Hst pairwise commute. We completely settle this problem by characterizing the full set of powers preserving positivity in this setting. Our characterizations generalize previous results by FitzGerald-Horn, Bhatia-Elsner, and Hiai from scalars to arbitrary block size, and in particular, generalize the Schur Product Theorem. Finally, a natural and unifying framework for studying the cases where the blocks Hst are diagonalizable consists of replacing real powers by general characters of the complex plane. We thus classify such characters, and generalize our results to this more general setting. In the course of our work, given β ∈ Z, we provide lower and upper bounds for the threshold power α > 0 above which the complex characters z = re iθ → r α e iβθ preserve positivity when applied entrywise to Hermitian positive semidefinite matrices. In particular, we completely resolve the n = 3 case of a question raised in 2001 by Xingzhi Zhan. As an application of our results, we also extend previous work by de Pillis [Duke Math. J. 36] by classifying the characters K of the complex plane for which the map (Hst) n s,t=1 → (K(tr(Hst))) n s,t=1 preserves Loewner positivity.

show abstract

The Schur Product Theorem in the Block Case

Cited by 3 publications

References 5 publications

Role Of Partial Transpose And Generalized Choi Maps In Quantum Dynamical Semigroups Involving Separable And Entangled States

Role Of Partial Transpose And Generalized Choi Maps In Quantum Dynamical Semigroups Involving Separable And Entangled States

Positive semidefinite 3 x 3 block matrices

On fractional Hadamard powers of positive block matrices

Contact Info

Product

Resources

About