22 papers · ranked by Valyu relevance
Ted Hurley
Basic matrices are defined which provide unique building blocks for the class of normal matrices which include the classes of unitary and Hermitian matrices. Unique builders for quantum logic gates are hence derived as a quantum logic gates is represented by, or is said to be, a unitary matrix. An efficient algorithm…
Cara D. Brooks, Alberto A. Condori
Given a normal matrix A and an arbitrary square matrix B (not necessarily of the same size), what relationships between A and B, if any, guarantee that B is also a normal matrix? We provide an answer to this question in terms of pseudospectra and norm behavior. In doing so, we prove that a certain distance formula…
Haoze He, Daniel Kressner
We present and analyze a simple numerical method that diagonalizes a complex normal matrix A by diagonalizing the Hermitian matrix obtained from a random linear combination of the Hermitian and skew-Hermitian parts of A.
Jean-Christophe Bourin, Eunyoung Lee
Let |Z|sym = (|Z ∗ | + |Z|)/2 denote the symetric modulus of Z ∈ Mn. The triangle inequality for this modulus and the operator norm ∥ · ∥ ∞ fails : Teng Zhang recently pointed out a simple example in M 2 with ∥|A + B|sym∥ ∞ > ∥|A|sym∥ ∞ + ∥|B|sym∥∞. However, for every symmetric (unitarily invariant) norm and any family…
Jia Zhao, Jieming Zhang
Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$J \in{\mathbb{R}}^{n\times n}$\end{document} J ∈ R n × n be a normal matrix such that…
Yi Shen, Lin Chen
We investigate the distillability problem in quantum information in $ℂd⊗ℂd$. One case of the problem has been reduced to proving a matrix inequality when $d=4$. We investigate the inequality for three families of non-normal matrices. We prove the inequality for the first two families with $d=4$ and for the third family…
Jose Divasón, René Thiemann
This work presents formal correctness proofs in Isabelle/HOL of algorithms to transform a matrix into Smith normal form, a canonical matrix form, in a general setting: the algorithms are written in an abstract form and parameterized by very few simple operations. We formally show their soundness provided the operations…
Ryan D. Wasson, Hugo J. Woerdeman
An n × n matrix A has a normal defect of k if there exists an (n + k) × (n + k) normal matrix Aext with A as a leading principal submatrix and k minimal. In this paper we compute the normal defect of a special class of 4 × 4 matrices, namely matrices whose only nonzero entries lie on the superdiagonal, and we provide…
Sergey V. Chermnykh
We study the matrix representation of Poincaré normalization using the Carleman linearization technique for non-autonomous differential systems with quasi-periodic coefficients. We provide a rigorous proof of the validity of the matrix representation of the normalization and obtain a recursive algorithm for computing…
D. S. Shirokov
We introduce the notion of rank of multivector in Clifford geometric algebras of arbitrary dimension without using the corresponding matrix representations and using only geometric algebra operations. We use the concepts of characteristic polynomial in geometric algebras and the method of SVD. The results can be used…
Patrick Otto Ludl
Most algorithms constructing bases of finite-dimensional vector spaces return basis vectors which, apart from orthogonality, do not show any special properties. While every basis is sufficient to define the vector space, not all bases are equally suited to unravel properties of the problem to be solved. In this paper a…
Shane Barratt
In this note, we define a Gaussian probability distribution over matrices. We prove some useful properties of this distribution, namely, the fact that marginalization, conditioning, and affine transformations preserve the matrix Gaussian distribution. We also derive useful results regarding the expected value of…
Authors not listed
This article is the second in a two-part tutorial review on electronic spin-dependent dynamics. In Part I, we presented the fundamental theory within the adiabatic Born– Huang framework that describes the interaction between nuclear motion and the elec- tronic (spin and spatial) degrees of freedom. In particular, we…
Victor P. Andreev, Gang Liu, Jarcy Zee, Lisa Henn + 2 more
Biological, ecological, social, and technological systems are complex structures with multiple interacting parts, often represented by networks. Correlation matrices describing interdependency of the variables in such structures provide key information for comparison and classification of such systems. Classification…
Sivan Leviyang
Modularity based clustering was introduced in the network literature for community detection and is now commonly applied to single cell RNA-seq (scRNAseq) datasets for cell type identification. Modularity clustering depends on a resolution parameter, which implicitly determines the number of clusters inferred, but no…
Evan D. Gorman, Manuel E. Lladser
Ultrametric matrices have a rich structure that is not apparent from their definition. Notably, the subclass of strictly ultrametric matrices are covariance matrices of certain weighted rooted binary trees. In applications, these matrices can be large and dense, making them difficult to store and handle. In this…
Chen Qu, Paul Houston, Qi Yu, Riccardo Conte + 3 more
Hamiltonian matrices in electronic and nuclear contexts are highly compute-intensive to calculate, mainly due to the cost for the potential matrix. Typically these matrices contain many off-diagonal elements that are orders of magnitude smaller than diagonal elements. We illustrate that here for vibrational H-matrices…
Alessandro Soncini, MATTEO PICCARDO
We present a non-orthogonal fragment ab initio methodology for the calculation of crystal field energy levels and magnetic properties in lanthanide complexes, implementing a systematic description of non-covalent contributions to metal-ligand bonding. The approach has two steps. In the first step, appropriate ab initio…
Authors not listed
Most advances in electronic spin-dependent non-adiabatic dynamics focus on refining the underlying dynamics methods. In contrast, this work considers an improved description of spin-orbit coupling by explicitly accounting for its relativistic origins. To this end, we extend a standard one-electron triatomic…
Bryan A. Dawkins, Trang T. Le, Brett A. McKinney
The performance of nearest-neighbor feature selection and prediction methods depends on the metric for computing neighborhoods and the distribution properties of the underlying data. The effects of the distribution and metric, as well as the presence of correlation and interactions, are reflected in the expected…
Jonas Vester, Jógvan Magnus Haugaard Olsen
The partial Hessian approximation is often used to perform vibrational analysis of large QM/MM systems where the high computational cost of calculating the full Hessian is impractical. Here, we investigate the accuracy and applicability of the partial Hessian vibrational analysis (PHVA) approach as it is typically used…
Hsing-Ta Chen, Junhan Chen, Vale Cofer-Shabica, Zeyu Zhou + 4 more
We present an efficient set of methods for propagating excited-state dynamics involving a large number of electronic states based on a CIS electronic state overlap scheme. Specifically, (i) following Head-Gordon et al, we implement an exact evaluation of the overlap of singly-excited electronic states at different…