The inverse eigenvalue problem for a Hermitian reflexive matrix and the optimization problem

The inverse eigenvalue problem and the associated optimal approximation problem for Hermitian reflexive matrices with respect to a normal {k+1}-potent matrix are considered. First, we study the existence of the solutions of the associated inverse eigenvalue problem and present an explicit form for them. Then, when such a solution exists, an expression for the solution to the corresponding optimal approximation problem is obtained

Inverse eigenvalue problem for normal J-hamiltonian matrices

A complex square matrix A is called J-hamiltonian if AJ is hermitian where J is a normal real matrix such that J^2=−I_n. In this paper we solve the problem of finding J-hamiltonian normal solutions for the inverse eigenvalue problem

Inverse eigenvalue problems of tridiagonal symmetric matrices and tridiagonal bisymmetric matrices

AbstractThe problem of generating a matrix A with specified eigenpairs, where A is a tridiagonal symmetric matrix, is presented. A general expression of such a matrix is provided, and the set of such matrices is denoted by SE. Moreover, the corresponding least-squares problem under spectral constraint is considered when the set SE is empty, and the corresponding solution set is denoted by SL. The best approximation problem associated with SE(SL) is discussed, that is: to find the nearest matrix  in SE(SL) to a given matrix. The existence and uniqueness of the best approximation are proved and the expression of this nearest matrix is provided. At the same time, we also discuss similar problems when A is a tridiagonal bisymmetric matrix

The Least Squares Hermitian (Anti)reflexive Solution with the Least Norm to Matrix Equation A

For a given generalized reflection matrix J, that is, JH=J, J2=I, where JH is the conjugate transpose matrix of J, a matrix A∈Cn×n is called a Hermitian (anti)reflexive matrix with respect to J if AH=A and A=±JAJ. By using the Kronecker product, we derive the explicit expression of least squares Hermitian (anti)reflexive solution with the least norm to matrix equation AXB=C over complex field

Duality for open fermion systems: energy-dependent weak coupling and quantum master equations

Open fermion systems with energy-independent bilinear coupling to a fermionic environment have been shown to obey a general duality relation [Phys. Rev. B 93, 81411 (2016)] which allows for a drastic simplification of time-evolution calculations. In the weak-coupling limit, such a system can be associated with a unique dual physical system in which all energies are inverted, in particular the internal interaction. This paper generalizes this fermionic duality in two ways: we allow for weak coupling with arbitrary energy dependence and describe both occupations and coherences coupled by a quantum master equation for the density operator. We also show that whenever generalized detailed balance holds (Kolmogorov criterion), the stationary probabilities for the dual system can be expressed explicitly in terms of the stationary recurrence times of the original system, even at large bias. We illustrate the generalized duality by a detailed analysis of the rate equation for a quantum dot with strong onsite Coulomb repulsion, going beyond the commonly assumed wideband limit. We present predictions for (i) the decay rates for transient charge and heat currents after a gate-voltage quench and (ii) the thermoelectric linear response coefficients in the stationary limit. We show that even for pronouncedly energy-dependent coupling, all nontrivial parameter dependence in these problems is entirely captured by just two well-understood stationary variables, the average charge of the system and of the dual system. Remarkably, it is the latter that often dictates the most striking features of the measurable quantities (e.g., positions of resonances), underscoring the importance of the dual system for understanding the actual one.Comment: 25 pages + 2 pages appendix + 2 pages references, 7 figures. To be submitted to Phys. Rev.