Semidefinite Matrices Research Articles

Parallel sum

In this paper, we will consider parallel summable operators on Hilbert space. Operators A and B are said to be parallel summable if and . We will prove that the parallel sum can be represented as A: B = A(A + B)† B without additional assumption of the closedness of the range of the operator A + B. Furthermore, we will derive the equalities C (A : B) = C A : C B and (A : B) : C = A : (B : C ) under weaker conditions than the ones represented in [X. Tian, S. Wang, C. Deng, On parallel sum of operators, Linear Algebra Appl. 603 (2020) 57–83] and [W. Luo, C. Song, Q. Xu, The parallel sum for adjointable operators on Hilbert C* -modules, Acta Math. Sin. 62 (2019) 541–552]. Finally, we will extend some recent result for Hermitian positive semi-definite matrices to bounded linear operators.

Generalized matrix functions and some refinement inequalities

In this short paper, we provide some refinement inequalities on generalized matrix functions. In particular, permanent inequalities concerning doubly stochastic positive semidefinite matrices are also included.

Quantum state estimation based on deep learning

Abstract We used deep learning techniques to construct various models for reconstructing quantum states from a given set of coincidence measurements. Through simulations, we demonstrated that our approach generates functionally equivalent reconstructed states for a wide range of pure and mixed input states. Compared to traditional methods, our system offers the advantage of faster speed. Additionally, by training our system with measurement results containing simulated noise sources, we could show a significant improvement in average fidelity compared to typical reconstruction methods. We also found that constraining the variational manifold to physical states, i.e., positive semi-definite density matrices, greatly enhances the quality of the reconstructed states in the presence of experimental imperfections and noise. Finally, we validated the correctness and superiority of our model by using data generated on IBMQ, a real quantum computer.

Some remarks on permanental dominance conjecture

In this paper we provide an identity between the determinant and other generalized matrix functions, and give a criterion for positive semi-definite matrices to satisfy the permanental dominance conjecture. As a consequence, infinitely many classes of positive semi-definite matrices satisfying the conjecture (does not depend on groups or characters) are provided by generating from any positive semi-definite matrix having no zero in the first column.

Deep neural networks as variational solutions for correlated open quantum systems

In this work we apply deep neural networks to find the non-equilibrium steady state solution to correlated open quantum many-body systems. Motivated by the ongoing search to find more powerful representations of (mixed) quantum states, we design a simple prototypical convolutional neural network and show that parametrizing the density matrix directly with more powerful models can yield better variational ansatz functions and improve upon results reached by neural density operator based on the restricted Boltzmann machine. Hereby we give up the explicit restriction to positive semi-definite density matrices. However, this is fulfilled again to good approximation by optimizing the parameters. The great advantage of this approach is that it opens up the possibility of exploring more complex network architectures that can be tailored to specific physical properties. We show how translation invariance can be enforced effortlessly and reach better results with fewer parameters. We present results for the dissipative one-dimensional transverse-field Ising model and a two-dimensional dissipative Heisenberg model compared to exact values.

Measuring Sphericity in Positive Semi-Definite Matrices

Measuring Sphericity in Positive Semi-Definite Matrices

On the matrix Heinz means and Rényi divergences

Bhatia, Lim and Yamazaki obtained results on the norm minimality of the Kubo-Ando Heinz means of positive semidefinite matrices. Namely, they proved that | | | A ♯ α B + A ♯ 1 − α B | | | ≤ | | | A α B 1 − α + A 1 − α B α | | | , for α ∈ ( 0 , 1 ) , in the case of the Schatten 1-norm or 2-norm. In this article, we refine these results via the matrix function in the definition of the Rényi divergence. We show that four different types of inequalities hold, under certain conditions, for Schatten 1- and 2-norms.

Disjoint sections of positive semidefinite matrices and their applications in linear statistical models

Abstract Given matrices A A and B B of the same order, A A is called a section of B B if R ( A ) ∩ R ( B − A ) = { 0 } {\mathscr{R}}\left(A)\cap {\mathscr{R}}\left(B-A)=\left\{0\right\} and R ( A T ) ∩ R ( ( B − A ) T ) = { 0 } {\mathscr{R}}\left({A}^{T})\cap {\mathscr{R}}\left({\left(B-A)}^{T})=\left\{0\right\} , where R ( . ) {\mathscr{R}}\left(.) denotes the range (column space) of a matrix argument and the superscript T T stands for the transpose of a matrix. A matrix G G is called positive semidefinite (p.s.d.) if x T G x {x}^{T}Gx is nonnegative for all real vector x x . However, we refer to a symmetric p.s.d. matrix simply as a p.s.d. matrix as the applications discussed in this article are concerned with only symmetric p.s.d. matrices. An n × n n\times n p.s.d. matrix G G admits of a symmetric section G X {G}_{X} of G G with regard to an n × k n\times k matrix X X such that R ( G X ) = R ( G ) ∩ R ( X ) {\mathscr{R}}\left({G}_{X})={\mathscr{R}}\left(G)\cap {\mathscr{R}}\left(X) . In this article, sections of the type G X {G}_{X} are used in minimization of quadratic functions under linear constraints and splitting of vector random variables into uncorrelated vector random variables. In the general Gauss-Markoff model, y = X β + ε y=X\beta +\varepsilon , with design matrix X X and a singular covariance matrix σ 2 G {\sigma }^{2}G of ε \varepsilon , y y is decomposed into four uncorrelated vector random variables as y = M 1 y + M 2 y + M 3 y + M 4 y y={M}_{1}y+{M}_{2}y+{M}_{3}y+{M}_{4}y , where M i , i = 1 , 2 , 3 {M}_{i},i=1,2,3 consist of sections of G G and X X T X{X}^{T} , and M 4 {M}_{4} is a matrix whose row space is the null space of G + X X T G+X{X}^{T} .

Permanents of block matrices

In this paper, we provide a formula to compute the permanent of block matrices depending on entries of each block. As a consequence, a generalized Lieb permanent inequality on positive semi-definite block matrices is given.

An SDP relaxation in the complex domain for the efficient coordination of BESS and DGs in single-phase distribution grids while considering reactive power capabilities

This research proposes an efficient energy management system design for medium-voltage distribution networks from a perspective of convex optimization. The exact nonlinear programming problem that represents the operation of distributed energy resources (DERs), such as batteries and photovoltaic sources, is approximated as an equivalent semi-definite programming (SDP) problem. This SDP approximation is achieved in the complex domain using Hermitian matrices, which are the equivalent of positive semi-definite matrices in the real number domain. Two test feeders, one with 27 nodes and the other with 33 nodes, are modified to match the characteristics of Colombian rural and urban networks. The 27-bus grid is adapted to accommodate the average demand and solar generation conditions of the municipality of Capurganá, while those of the city of Medellín are used for the 33-bus grid. One of the main contributions of this research is that it considers a DER operation with variable power factors, which allows for significant improvements compared to the traditional use of a unitary power factor scheme. Numerical comparisons are conducted using metaheuristic optimizers, including parallel versions of the particle swarm optimizer, the vortex search algorithm, and the continuous version of the genetic algorithm. These comparisons demonstrate the effectiveness of the proposed SDP approximation. All simulations are carried out in the MATLAB software (version 2022b), using the disciplined convex optimizer (CVX) and the SDPT3 and SEDUMI solvers.

Further norm inequalities for positive semidefinite matrices

Further norm inequalities for positive semidefinite matrices

A GenEO Domain Decomposition method for Saddle Point problems

We introduce an adaptive element-based domain decomposition (DD) method for solving saddle point problems defined as a block two by two matrix. The algorithm does not require any knowledge of the constrained space. We assume that all sub matrices are sparse and that the diagonal blocks are spectrally equivalent to a sum of positive semi definite matrices. The latter assumption enables the design of adaptive coarse space for DD methods that extends the GenEO theory (Spillane et al., 2014) to saddle point problems. Numerical results on three dimensional elasticity problems for steel-rubber structures discretized by a finite element with continuous pressure are shown for up to one billion degrees of freedom.

A Predictor-Corrector Algorithm for Semidefinite Programming that Uses the Factor Width Cone

AbstractWe propose an interior point method (IPM) for solving semidefinite programming problems (SDPs). The standard interior point algorithms used to solve SDPs work in the space of positive semidefinite matrices. Contrary to that the proposed algorithm works in the cone of matrices of constant factor width. We prove global convergence and provide a complexity analysis. Our work is inspired by a series of papers by Ahmadi, Dash, Majumdar and Hall, and builds upon a recent preprint by Roig-Solvas and Sznaier [arXiv:2202.12374, 2022].

Short note on some geometric inequalities derived from matrix inequalities

Using the connection between ellipsoids and positive semidefinite matrices we provide alternative proofs to some recently proven inequalities concerning the volume of L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document} zonoids as consequences of classical inequalities for matrices.

Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices

Abstract This article introduces new Young-type inequalities, leveraging the Kantorovich constant, by refining the original inequality. In addition, we present a range of norm-based inequalities applicable to positive semidefinite matrices, such as the Hilbert-Schmidt norm and the trace norm. The importance of these results lies in their dual significance: they hold inherent value on their own, and they also extend and build upon numerous established results within the existing literature.

Partial trace inequalities for partial transpose of positive semidefinite block matrices

Partial trace inequalities for partial transpose of positive semidefinite block matrices

Inertia of partial transpose of positive semidefinite matrices

We show that the partial transpose of 9×9 positive semidefinite matrices do not have inertia (4,1,4) and (3,2,4) . It solves an open problem in ‘LINEAR AND MULTILINEAR ALGEBRA. Changchun Feng et al, 2022’. We apply our results to construct some inertia, as well as present the list of all possible inertia of partial transpose of 12×12 positive semidefinite matrices.

Modified CRI iteration methods for complex symmetric indefinite linear systems

This work investigates the iterative solution of complex symmetric linear systems with indefinite matrix term. Based on a technical equivalent reformulation of the original indefinite systems, an efficient stationary splitting iteration method is established by combining with its real and imaginary parts. In the view of the structure, the new method can be viewed as a modification of the combination method of the real and imaginary parts (CRI) in [Wang & Lu, J Comput Appl Math. 2017;325:188–197], which is originally designed for solving complex symmetric linear systems with positive semi-definite coefficient matrices. Despite that it is devoted to more difficult indefinite problems, the new method keeps the merits of unconditional convergence and parameter free implementation of the original method. Moreover, Chebyshev polynomial acceleration is considered to further improve the convergence rate of the new method. Numerical experiments on a set of model problems are tested to illustrate the efficiency of the proposed methods compared with some existing ones.

Efficient Identification of Error-in-Variables Switched Systems Using a Riemannian Embedding

This paper considers the problem of error in variables identification for switched affine models. Since it is well known that this problem is generically NP hard, several relaxations have been proposed in the literature. However, while these approaches work well for low dimensional systems with few subsystems, they scale poorly with both the number of subsystems and their memory. To address this difficulty, we propose a computationally efficient alternative, based on embedding the data in the manifold of positive semidefinite matrices, and using a manifold metric there to perform the identification. Our main result shows that, under dwell-time assumptions, the proposed algorithm is convergent, in the sense that it is guaranteed to identify the system for suitably low noise. In scenarios with larger noise levels, we provide experimental results showing that the proposed method outperforms existing ones. The paper concludes by illustrating these results with academic examples and a non-trivial application: action video segmentation.

Singular Value and Matrix Norm Inequalities between Positive Semidefinite Matrices and Their Blocks

In this paper, we obtain some inequalities involving positive semidefinite 2×2 block matrices and their blocks.