How to Calculate the Exponential of Matrices

具有幺模谱的矩阵的伪谱所导致的繁荣与萧条周期

Boom and bust cycles due to pseudospectra of matrices with unimodular spectra

本研究揭示了特征值在单位圆上的非正规矩阵高次幂动力学的反直觉行为：无限大矩阵可指数增长，有限矩阵呈增长-衰减周期，该效应由谱与伪谱的差异决定，且存在于多种物理与数学模型中。

Simply Exponential Approximation of the Permanent of Positive Semidefinite Matrices

We design a deterministic polynomial time $c^n$ approximation algorithm for the permanent of positive semidefinite matrices where $c=e^{γ+1}\simeq 4.84$. We write a natural convex relaxation and show that its optimum solution gives a $c^n$ approximation of the permanent. We further show that this factor is asymptotically tight by constructing a family of positive semidefinite matrices.

关于极限周期雅可比矩阵的A.C.谱的注记

A note on the A.C. spectrum of limit-periodic Jacobi matrices

该研究针对弱指数类EC(η)的极限周期雅可比矩阵，通过建立谱齐次性、传递无反射性并应用Poltoratski与Remling定理，证明η足够大时其具有纯绝对连续谱型。

Explicit Inverse Confluent Vandermonde Matrices with Applications to Exponential Quantum Operators

The Cayley-Hamilton problem of expressing functions of matrices in terms of only their eigenvalues is well-known to simplify to finding the inverse of the confluent Vandermonde matrix. Here, we give a highly compact formula for the inverse of any matrix, and apply it to the confluent Vandermonde matrix, achieving in a single equation what has only been achieved by long iterative algorithms until now. As a prime application, we use this result to get a simple formula for explicit exponential operators in terms of only their eigenvalues, with an emphasis on application to finite discrete quantum…

All Moments of the Uniform Ensemble of Quantum Density Matrices

Given a uniform ensemble of quantum density matrices $ρ$, it is useful to calculate the mean value over this ensemble of a product of entries of $ρ$. We show how to calculate such moments in this paper. The answer involves well known results from Group Representation Theory and Random Matrix Theory. This quantum problem has a well known classical counterpart: given a uniform ensemble of probability distributions $P=(P_1, P_2, ..., P_N)$ where the $P_j$ are non-negative reals that sum to one, calculate the mean value over this probability simplex of products of $P$ components. The answer to the…

全秩和低秩指数欧拉积分算子用于林布拉德方程

Full- and low-rank exponential Euler integrators for the Lindblad equation

本文提出全秩和低秩指数欧拉积分算子，用于近似林布拉德方程，保证正定性和迹不变性，理论结果提供误差估计，数值实验展示方法有效性。