A Singular Value Inequality Related to a Linear Map

2016 ◽  
Vol 31 ◽  
pp. 120-124 ◽  
Author(s):  
Minghua Lin
Keyword(s):  
Linear Algebra ◽  
Positive Semidefinite ◽  
Singular Value ◽  
Linear Map

If $\begin{bmatrix}A & X \\ X^* & B\end{bmatrix}$ is positive semidefinite with each block $n\times n$, we prove that $$2s_j\Big(\Phi(X)\Big)\le s_j\Big(\Phi(A+B)\Big), \qquad j=1, \ldots, n,$$ where $\Phi: X\mapsto X+(\tr X)I$ and $s_j(\cdot)$ means the $j$-th largest singular value. This confirms a conjecture of the author in [Linear Algebra Appl. 459 (2014) 404-410].

scholarly journals Some singular value inequalities related to linear maps

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3705-3709
Author(s):  
Junjian Yang ◽  
Linzhang Lu ◽  
Zhen Chen
Keyword(s):  
Positive Semidefinite ◽  
Singular Value ◽  
Linear Map ◽  
Linear Maps ◽  
Alternative Approach

If (A X X* B) ? M2(Mn) is positive semidefinite, Lin [7] conjectured that 2sj(?(X))?sj(?(A)+ (B)), j=1,..., n, and sj(?(X))? sj(?(A)#?(B)), j = 1,..., n, where the linear map ?:X 7 ? 2tr(X)In-X and sj(?) means the j-th largest singular value. In this paper, we reprove that (?(A) ?(X) ?(X*)?(B)) is PPT by using an alternative approach and prove the above singular value inequalities hold for the linear map ?1 : X ?(2n + 1)tr(X)In-X.

scholarly journals Some algorithms for maximum volume and cross approximation of symmetric semidefinite matrices

2021 ◽  
Author(s):  
Stefano Massei
Keyword(s):  
Computer Science ◽  
Recent Work ◽  
Linear Algebra ◽  
Optimization Problems ◽  
Approximation Error ◽  
Positive Semidefinite ◽  
Numerical Linear Algebra ◽  
Maximum Volume ◽  
Deterministic Algorithms ◽  
Principal Submatrix

AbstractVarious applications in numerical linear algebra and computer science are related to selecting the $$r\times r$$ r × r submatrix of maximum volume contained in a given matrix $$A\in \mathbb R^{n\times n}$$ A ∈ R n × n . We propose a new greedy algorithm of cost $$\mathcal O(n)$$ O ( n ) , for the case A symmetric positive semidefinite (SPSD) and we discuss its extension to related optimization problems such as the maximum ratio of volumes. In the second part of the paper we prove that any SPSD matrix admits a cross approximation built on a principal submatrix whose approximation error is bounded by $$(r+1)$$ ( r + 1 ) times the error of the best rank r approximation in the nuclear norm. In the spirit of recent work by Cortinovis and Kressner we derive some deterministic algorithms, which are capable to retrieve a quasi optimal cross approximation with cost $$\mathcal O(n^3)$$ O ( n 3 ) .

Singular Value and Norm Inequalities Associated with 2 x 2 Positive Semidefinite Block Matrices

2017 ◽  
Vol 32 ◽  
pp. 116-124 ◽  
Author(s):  
Aliaa Burqan ◽  
Fuad Kittaneh
Keyword(s):  
Positive Semidefinite ◽  
Singular Value ◽  
Norm Inequalities ◽  
Block Matrices

This paper aims to give singular value and norm inequalities associated with $2\times 2$ positive semidefinite block matrices.

Inequalities for relative operator entropies

2014 ◽  
Vol 27 ◽  
Author(s):  
Pawel Kluza ◽  
Marek Niezgoda
Keyword(s):  
Linear Algebra ◽  
Operator Inequalities ◽  
Linear Map ◽  
Positive Linear Map ◽  
Kantorovich Constant ◽  
Reverse Inequalities

In this paper, operator inequalities are provided for operator entropies transformed by a strictly positive linear map. Some results by Furuichi et al. [S. Furuichi, K. Yanagi, and K. Kuriyama. A note on operator inequalities of Tsallis relative operator entropy. Linear Algebra Appl., 407:19–31, 2005.], Furuta [T. Furuta. Two reverse inequalities associated with Tsallis relative operator entropy via generalized Kantorovich constant and their applications. Linear Algebra Appl., 412:526–537, 2006.], and Zou [L. Zou. Operator inequalities associated with Tsallis relative operator entropy. Math. Inequal. Appl., 18:401–406, 2015.] are extended. In particular, the obtained inequalities are specified for relative operator entropy and Tsallis relative operator entropy. In addition, some bounds for generalized relative operator entropy are established.

scholarly journals Using SVD on Clusters to Improve Precision of Interdocument Similarity Measure

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Wen Zhang ◽  
Fan Xiao ◽  
Bin Li ◽  
Siguang Zhang
Keyword(s):  
Singular Value Decomposition ◽  
Linear Algebra ◽  
Similarity Measure ◽  
Latent Semantic Indexing ◽  
Singular Value ◽  
Semantic Indexing ◽  
Discriminative Power ◽  
Good Representative ◽  
Value Decomposition ◽  
Dimension Expansion

Recently, LSI (Latent Semantic Indexing) based on SVD (Singular Value Decomposition) is proposed to overcome the problems of polysemy and homonym in traditional lexical matching. However, it is usually criticized as with low discriminative power for representing documents although it has been validated as with good representative quality. In this paper, SVD on clusters is proposed to improve the discriminative power of LSI. The contribution of this paper is three manifolds. Firstly, we make a survey of existing linear algebra methods for LSI, including both SVD based methods and non-SVD based methods. Secondly, we propose SVD on clusters for LSI and theoretically explain that dimension expansion of document vectors and dimension projection using SVD are the two manipulations involved in SVD on clusters. Moreover, we develop updating processes to fold in new documents and terms in a decomposed matrix by SVD on clusters. Thirdly, two corpora, a Chinese corpus and an English corpus, are used to evaluate the performances of the proposed methods. Experiments demonstrate that, to some extent, SVD on clusters can improve the precision of interdocument similarity measure in comparison with other SVD based LSI methods.

scholarly journals An elementary proof of Chollet’s permanent conjecture for 4 × 4 real matrices

2021 ◽  
Vol 9 (1) ◽  
pp. 83-102
Author(s):  
George Hutchinson
Keyword(s):  
Linear Algebra ◽  
Elementary Proof ◽  
Positive Semidefinite ◽  
Real Matrices

Abstract A proof of the statement per(A ∘ B) ≤ per(A)per(B) is given for 4 × 4 positive semidefinite real matrices. The proof uses only elementary linear algebra and a rather lengthy series of simple inequalities.

scholarly journals SPN Graphs

2019 ◽  
Vol 35 ◽  
pp. 376-386
Author(s):  
Leslie Hogben ◽  
Naomi Shaked-Monderer
Keyword(s):  
Linear Algebra ◽  
Nonnegative Matrix ◽  
Positive Semidefinite ◽  
Simple Graph ◽  
Subdivision Graph ◽  
Copositive Matrix

A simple graph G is an SPN graph if every copositive matrix having graph G is the sum of a positive semidefinite and nonnegative matrix. SPN graphs were introduced in [N. Shaked-Monderer. SPN graphs: When copositive = SPN. Linear Algebra Appl., 509:82{113, 2016.], where it was conjectured that the complete subdivision graph of K4 is an SPN graph. This conjecture is disproved, which in conjunction with results in the Shaked-Monderer paper show that a subdivision of K_4 is a SPN graph if and only if at most one edge is subdivided. It is conjectured that a graph is an SPN graph if and only if it does not have an F_5 minor, where F_5 is the fan on five vertices. To establish that the complete subdivision graph of K_4 is not an SPN graph, rank-1 completions are introduced and graphs that are rank-1 completable are characterized.

scholarly journals The Non-m-Positive Dimension of a Positive Linear Map

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 172 ◽  
Author(s):  
Nathaniel Johnston ◽  
Benjamin Lovitz ◽  
Daniel Puzzuoli
Keyword(s):  
Schmidt Number ◽  
Positive Semidefinite ◽  
Positive Dimension ◽  
Linear Map ◽  
Positive Partial Transpose ◽  
Positive Maps ◽  
Negative Eigenvalues ◽  
Entanglement Witnesses ◽  
Positive Linear Map ◽  
Partial Transpose

We introduce a property of a matrix-valued linear map Φ that we call its ``non-m-positive dimension'' (or ``non-mP dimension'' for short), which measures how large a subspace can be if every quantum state supported on the subspace is non-positive under the action of Im⊗Φ. Equivalently, the non-mP dimension of Φ tells us the maximal number of negative eigenvalues that the adjoint map Im⊗Φ∗ can produce from a positive semidefinite input. We explore the basic properties of this quantity and show that it can be thought of as a measure of how good Φ is at detecting entanglement in quantum states. We derive non-trivial bounds for this quantity for some well-known positive maps of interest, including the transpose map, reduction map, Choi map, and Breuer--Hall map. We also extend some of our results to the case of higher Schmidt number as well as the multipartite case. In particular, we construct the largest possible multipartite subspace with the property that every state supported on that subspace has non-positive partial transpose across at least one bipartite cut, and we use our results to construct multipartite decomposable entanglement witnesses with the maximum number of negative eigenvalues.

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