Related papers: The eigenvector-eigenvalue identity for the quater…
The history of research on eigenvalue problems is rich with many outstanding contributions. Nonetheless, the rapidly increasing size of data sets requires new algorithms for old problems in the context of extremely large matrix dimensions.…
We present a prescription for forming matrices with specified eigenvalues and known eigenvectors. With this method, we can form Hermitian, anti-Hermitian, symmetric and general matrices with arbitrary eigenvalues. In addition we propose an…
We derive out a complete series expression of Hamiltonian eigenvalues without any approximation and cut in the general quantum systems based on Wang's formal framework \cite{wang1}. In particular, we then propose a calculating approach of…
This paper presents a method for computing eigenvalues and eigenvectors for some types of nonlinear eigenvalue problems. The main idea is to approximate the functions involved in the eigenvalue problem by rational functions and then apply a…
Improving upon results of Rudelson and Vershynin, we establish delocalization bounds for eigenvectors of independent-entry random matrices. In particular, we show that with high probability every eigenvector is delocalized, meaning any…
In order to guarantee the downloading quality requirements of users and improve the stability of data transmission in a BitTorrent-like peer-to-peer file sharing system, this article deals with eigenproblems of addition-min algebras. First,…
The eigenvalues of an arbitrary quaternionic matrix have a joint probability distribution function first derived by Ginibre. We show that there exists a mapping of this system onto a fermionic field theory and then use this mapping to…
Eigenvalue transformations appear ubiquitously in scientific computation, ranging from matrix polynomials to differential equations, and are beyond the reach of the quantum singular value transformation framework. In this work, we study the…
The diagonalization of matrices may be the top priority in the application of modern physics. In this paper, we numerically demonstrate that, for real symmetric random matrices with non-positive off-diagonal elements, a universal scaling…
We consider random matrices of the form $H = W + \lambda V$, $\lambda\in\mathbb{R}^+$, where $W$ is a real symmetric or complex Hermitian Wigner matrix of size $N$ and $V$ is a real bounded diagonal random matrix of size $N$ with i.i.d.\…
Eigenvalue distributions are important dynamical quantities in matrix models, and it is a challenging problem to derive them in tensor models. In this paper, we consider real symmetric order-three tensors with Gaussian distributions as the…
A methodology to analyze the properties of the first (largest) eigenvalue and its eigenvector is developed for large symmetric random sparse matrices utilizing the cavity method of statistical mechanics. Under a tree approximation, which is…
We propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highly ill-conditioned matrices. It is based on the {\it LDLT} decomposition and involves finding a $k \times k$ sub-matrix of the inverse of the…
This paper offers a review of numerical methods for computation of the eigenvalues of Hermitian matrices and the singular values of general and some classes of structured matrices. The focus is on the main principles behind the methods that…
Assume that the eigenvalues of a finite hermitian linear operator have been deduced accurately but the linear operator itself could not be determined with precision. Given a set of eigenvalues $\lambda$ and a hermitian matrix $M$, this…
We present a closed-form solution to Wahba's problem in the quaternion domain for the special case of two vector observations. Existing approaches, including Davenport's $q$-method, QUEST, Horn's method, and ESOQ algorithms, recover the…
We present an efficient method for estimating the eigenvalues of a Hamiltonian $H$ from the expectation values of the evolution operator for various times. For a given quantum state $\rho$, our method outputs a list of eigenvalue estimates…
Conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex Hermitian matrices with a bounded number of distinct eigenvalues. A minimal generating system of the vanishing ideal of degenerate…
This article considers the problem of designing adaption and optimisation techniques for training quantum learning machines. To this end, the division algebra of quaternions is used to derive an effective model for representing computation…
Fine-grained spectral properties of quantum Hamiltonians, including both eigenvalues and their multiplicities, provide useful information for characterizing many-body quantum systems as well as for understanding phenomena such as…