A New Trace Bound for a General Square Matrix Product

2007 ◽  
Vol 52 (2) ◽  
pp. 349-352 ◽  
Author(s):  
Jianzhou Liu ◽  
Lingli He
Keyword(s):  
Matrix Product ◽  
Square Matrix

A trace bound for a general square matrix product

2000 ◽  
Vol 45 (8) ◽  
pp. 1563-1569 ◽  
Author(s):  
Wei Xing ◽  
Qingling Zhang ◽  
Qiyi Wang
Keyword(s):  
Matrix Product ◽  
Square Matrix

scholarly journals A note on two-sided removal and cancellation properties associated with Hermitian matrix

2021 ◽  
Author(s):  
Yongge Tian
Keyword(s):  
Hermitian Matrix ◽  
Matrix Product ◽  
Conjugate Transpose ◽  
Square Matrix

Abstract A complex square matrix A is said to be Hermitian if A = A∗, the conjugate transpose of A. We prove that each of the two triple matrix product equalities AA∗A = A∗AA∗ and A3 = AA∗A implies that A is Hermitian by means of decompositions and determinants of matrices, which are named as two-sided removal and cancellation laws associated with a Hermitian matrix. We also present several general removal and cancellation laws as the extensions of the preceding two facts about Hermitian matrix.AMS classifications: 15A24, 15B57

Stochastic local operations and classical communication invariants via square matrix

Laser Physics ◽  
2019 ◽  
Vol 29 (2) ◽  
pp. 025203 ◽  
Author(s):  
Xinwei Zha ◽  
Irfan Ahmed ◽  
Da Zhang ◽  
Wen Feng ◽  
Yanpeng Zhang
Keyword(s):  
Classical Communication ◽  
Square Matrix

scholarly journals Numerical Analysis of Viscoelastic Rotating Beam with Variable Fractional Order Model Using Shifted Bernstein–Legendre Polynomial Collocation Algorithm

2021 ◽  
Vol 5 (1) ◽  
pp. 8
Author(s):  
Cundi Han ◽  
Yiming Chen ◽  
Da-Yan Liu ◽  
Driss Boutat
Keyword(s):  
Numerical Analysis ◽  
Fractional Order ◽  
Governing Equation ◽  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Matrix Product ◽  
Algebraic Equations ◽  
Rotating Beam ◽  
The Matrix ◽  
The Time Domain

This paper applies a numerical method of polynomial function approximation to the numerical analysis of variable fractional order viscoelastic rotating beam. First, the governing equation of the viscoelastic rotating beam is established based on the variable fractional model of the viscoelastic material. Second, shifted Bernstein polynomials and Legendre polynomials are used as basis functions to approximate the governing equation and the original equation is converted to matrix product form. Based on the configuration method, the matrix equation is further transformed into algebraic equations and numerical solutions of the governing equation are obtained directly in the time domain. Finally, the efficiency of the proposed algorithm is proved by analyzing the numerical solutions of the displacement of rotating beam under different loads.

scholarly journals Optimal sampling of dynamical large deviations via matrix product states

2021 ◽  
Vol 103 (6) ◽  
Author(s):  
Luke Causer ◽  
Mari Carmen Bañuls ◽  
Juan P. Garrahan
Keyword(s):  
Large Deviations ◽  
Matrix Product ◽  
Optimal Sampling ◽  
Matrix Product States

scholarly journals Matrix Product State Simulations of Non-Equilibrium Steady States and Transient Heat Flows in the Two-Bath Spin-Boson Model at Finite Temperatures

Entropy ◽  
2021 ◽  
Vol 23 (1) ◽  
pp. 77
Author(s):  
Angus J. Dunnett ◽  
Alex W. Chin
Keyword(s):  
Finite Temperature ◽  
Open Quantum Systems ◽  
Quantum Systems ◽  
Steady States ◽  
Matrix Product ◽  
Open Quantum ◽  
Matrix Product State ◽  
Wide Range ◽  
Boson Model ◽  
Non Equilibrium

Simulating the non-perturbative and non-Markovian dynamics of open quantum systems is a very challenging many body problem, due to the need to evolve both the system and its environments on an equal footing. Tensor network and matrix product states (MPS) have emerged as powerful tools for open system models, but the numerical resources required to treat finite-temperature environments grow extremely rapidly and limit their applications. In this study we use time-dependent variational evolution of MPS to explore the striking theory of Tamascelli et al. (Phys. Rev. Lett. 2019, 123, 090402.) that shows how finite-temperature open dynamics can be obtained from zero temperature, i.e., pure wave function, simulations. Using this approach, we produce a benchmark dataset for the dynamics of the Ohmic spin-boson model across a wide range of coupling strengths and temperatures, and also present a detailed analysis of the numerical costs of simulating non-equilibrium steady states, such as those emerging from the non-perturbative coupling of a qubit to baths at different temperatures. Despite ever-growing resource requirements, we find that converged non-perturbative results can be obtained, and we discuss a number of recent ideas and numerical techniques that should allow wide application of MPS to complex open quantum systems.

scholarly journals Algebraic conditions and the sparsity of spectrally arbitrary patterns

2021 ◽  
Vol 9 (1) ◽  
pp. 257-274
Author(s):  
Louis Deaett ◽  
Colin Garnett
Keyword(s):  
Major Goal ◽  
Algebraic Condition ◽  
Open Question ◽  
Algebraic Tool ◽  
General Method ◽  
Square Matrix ◽  
Spectrally Arbitrary

Abstract Given a square matrix A, replacing each of its nonzero entries with the symbol * gives its zero-nonzero pattern. Such a pattern is said to be spectrally arbitrary when it carries essentially no information about the eigenvalues of A. A longstanding open question concerns the smallest possible number of nonzero entries in an n × n spectrally arbitrary pattern. The Generalized 2n Conjecture states that, for a pattern that meets an appropriate irreducibility condition, this number is 2n. An example of Shitov shows that this irreducibility is essential; following his technique, we construct a smaller such example. We then develop an appropriate algebraic condition and apply it computationally to show that, for n ≤ 7, the conjecture does hold for ℝ, and that there are essentially only two possible counterexamples over ℂ. Examining these two patterns, we highlight the problem of determining whether or not either is in fact spectrally arbitrary over ℂ. A general method for making this determination for a pattern remains a major goal; we introduce an algebraic tool that may be helpful.

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