Hermitian Matrices Research Articles

The altered Hermite matrix: implications and ramifications

<p>Matrix theory is essential for addressing practical problems and executing computational tasks. Matrices related to Hermite polynomials are essential due to their applications in quantum mechanics, numerical analysis, probability, and signal processing. Their orthogonality, recurrence relations, and spectral properties make them a valuable tool for both theoretical research and practical applications. From a different perspective, we introduced a variant of the Hermite matrix that incorporates triple factorials and demonstrated that this matrix satisfies various properties. By utilizing effective matrix algebra techniques, various algebraic properties of this matrix have been determined, including the product formula, inverse matrix and eigenvalues. Additionally, we extended this matrix to a more generalized form and derived several identities.</p>

MATRIX FEJÉR AND LEVIN-STEČKIN INEQUALITIES

Fejér and Levin-Stečkin inequalities treat integrals of the product of convex functions with symmetric functions. The main goal of this article is to present possible matrix versions of these inequalities. In particular, majorization results are shown of Fejér type for both convex and log-convex functions. For the matrix Levin-Stečkin type, we present more rigorous results involving the partial Löewner ordering for Hermitian matrices. Further related results involving synchronous functions are presented, too.

Asymptotic expansion of matrix models in the multi-cut regime

Abstract We establish the asymptotic expansion in $\beta $ matrix models with a confining, off-critical potential in the regime where the support of the equilibrium measure is a finite union of segments. We first address the case where the filling fractions of these segments are fixed and show the existence of a $\frac {1}{N}$ expansion. We then study the asymptotics of the sum over the filling fractions to obtain the full asymptotic expansion for the initial problem in the multi-cut regime. In particular, we identify the fluctuations of the linear statistics and show that they are approximated in law by the sum of a Gaussian random variable and an independent Gaussian discrete random variable with oscillating center. Fluctuations of filling fractions are also described by an oscillating discrete Gaussian random variable. We apply our results to study the all-order small dispersion asymptotics of solutions of the Toda chain associated with the one Hermitian matrix model ( $\beta = 2$ ) as well as orthogonal ( $\beta = 1$ ) and skew-orthogonal ( $\beta = 4$ ) polynomials outside the bulk.

Perfect Hermitian matrix and dual quaternion Löwner partial orders

Perfect Hermitian matrix and dual quaternion Löwner partial orders

Homogenization of elliptic and parabolic equations with periodic coefficients in a bounded domain under the Neumann condition

Let $\mathcal{O}\subset\mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a selfadjoint matrix elliptic second-order differential operator $B_{N,\varepsilon}$, $0<\varepsilon\leqslant1$, with the Neumann boundary condition. The principal part of this operator is given in a factorized form. The operator involves first-order and zero-order terms. The coefficients of $B_{N,\varepsilon}$ are periodic and depend on $\mathbf{x}/\varepsilon$. We study the generalized resolvent $(B_{N,\varepsilon}-\zeta Q_0( {\cdot} /\varepsilon))^{-1}$, where $Q_0$ is a periodic bounded and positive definite matrix-valued function, and $\zeta$ is a complex-valued parameter. We obtain approximations for the generalized resolvent in the operator norm on $L_2(\mathcal{O};\mathbb{C}^n)$ and in the norm of operators acting from $L_2(\mathcal{O};\mathbb{C}^n)$ to the Sobolev space $H^1(\mathcal{O};\mathbb{C}^n)$ with two-parametric error estimates (with respect to $\varepsilon$ and $\zeta$). The results are applied to study the behaviour of the solutions of the initial boundary value problem with the Neumann condition for the parabolic equation $Q_0(\mathbf{x}/\varepsilon) \partial_t \mathbf{u}_\varepsilon(\mathbf{x},t) = -(B_{N,\varepsilon} \mathbf{u}_\varepsilon)(\mathbf{x},t)$ in a cylinder $\mathcal{O} \times (0,T)$, where $0<T \le \infty$.

Randic Skew - Hermitian Matrix and Randic Skew - Hermitian Energy of Mixed Middle Graphs

Let 〖SH(M〗_di) be the matrix of Skew-Hermitian adjacency and let M_dibe an asymmetrical middle graph. A latterly weighed Skew-Hermitian connection matric can be obtained as we proceed assign a Randic the amount as to each curve and boundary in 〖SH(M〗_di). In light of here, we as a species describe the Randic Skew-Hermitian matrix using these Skew-Hermitian adjacency matrix features. 〖R^*〗_SH (M_di )=(〖r^*〗_SH )_xy within a asymmetrical graph M_di Whereas (〖r^*〗_SH )_xy=((-i)/√(d_x d_y )) (i=√(-1) if (v_x,v_y )) is an arc of M_di, (〖r^*〗_SH )_xy=(i/√(d_x d_y ))(i=√(-1) if (v_x,v_y )) is an arc of M_di, (〖r^*〗_SH )_xy=((-1)/√(d_x d_y )) if (v_x,v_y ) is an undirected edge of M_di, and (〖r^*〗_SH )_xy=0 if of M_di. The main purpose of this study is to calculate the Randic Skew-Hermitian matrix of a asymmetrical middle graph's characteristic polynomial. Moreover, we provide boundaries on the applicable asymmetrical middle graph's Randic Skew-Hermitian energy. Finally, we provide some findings regarding the asymmetrical middle graphs' Randic energy with skew-Hermitian .

Exclusion principle between the physicochemical properties of complementary nucleobases and symmetry breaking in double-stranded DNA conformations

The information contained in DNA is encoded in a double helix formed by two antiparallel and complementary sequences of nucleotides, which are stabilized by hydrogen bonds. At physiological conditions, each nucleobase possesses three main characteristics; molecular structure, functional group, and orientation in the double helix. These properties produce steric constraints that allow only complementary interactions between nucleobase pairs to form double or triple hydrogen bonds. In this work, a mathematical model is proposed to describe the pairing interactions between nucleobases according to their physicochemical properties (structure, functional group and orientation). The results suggest that stable hydrogen bonds derived from nucleobase interactions can be represented by a set of 4 hermitian matrices and a Clifford Algebra Cl(4,0). Moreover, it is shown that the allowed interactions between pairs of nucleobases enable the formulation of an exclusion principle between the states of the observables. Finally, the mathematical representation of the hydrogen bonding between complementary nucleobases suggests a preferential orientation (chirality), which could be associated with the major groove in the double helix structure of B-DNA.

A fast algorithm for the Schrödinger equation in quaternionic quantum mechanics

The eigenvalue problem of a Hermitian quaternion matrix plays a crucial role in quaternion quantum mechanics because it is closely related to the solution of Schrödinger equation. In this paper, a fast algorithm is proposed for finding the eigenvalues and corresponding eigenvectors of a Hermitian quaternion matrix based on the real representation of a quaternion matrix as well as the special structure and properties of a Hermitian quaternion matrix. Numerical experiments demonstrate that, compared with the existing computational methods for the eigenvalue problem of a Hermitian quaternion matrix, the proposed method in this paper not only greatly improves the computational efficiency, but also achieves better experimental results in terms of the corresponding computational errors.

A universal variational quantum eigensolver for non-Hermitian systems

Many quantum algorithms are developed to evaluate eigenvalues for Hermitian matrices. However, few practical approach exists for the eigenanalysis of non-Hermintian ones, such as arising from modern power systems. The main difficulty lies in the fact that, as the eigenvector matrix of a general matrix can be non-unitary, solving a general eigenvalue problem is inherently incompatible with existing unitary-gate-based quantum methods. To fill this gap, this paper introduces a Variational Quantum Universal Eigensolver (VQUE), which is deployable on noisy intermediate scale quantum computers. Our new contributions include: (1) The first universal variational quantum algorithm capable of evaluating the eigenvalues of non-Hermitian matrices—Inspired by Schur’s triangularization theory, VQUE unitarizes the eigenvalue problem to a procedure of searching unitary transformation matrices via quantum devices; (2) A Quantum Process Snapshot technique is devised to make VQUE maintain the potential quantum advantage inherited from the original variational quantum eigensolver—With additional O(log2N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(log_{2}{N})$$\\end{document} quantum gates, this method efficiently identifies whether a unitary operator is triangular with respect to a given basis; (3) Successful deployment and validation of VQUE on a real noisy quantum computer, which demonstrates the algorithm’s feasibility. We also undertake a comprehensive parametric study to validate VQUE’s scalability, generality, and performance in realistic applications.

Universal eigenvalue statistics for dynamically defined matrices

We consider dynamically defined Hermitian matrices generated from orbits of the doubling map. We prove that their spectra fall into the GUE universality class from random matrix theory.

Some remarks on the norm upper bounds associated with the generalized polar decompositions of matrices

A correction, along with an improvement, is made on a recent joint work of the second author and the third author. An optimal perturbation bound is also clarified for certain 2×2 Hermitian matrices.

Some New Characterizations of a Hermitian Matrix and Their Applications

Some New Characterizations of a Hermitian Matrix and Their Applications

Hermitians in matrix algebras with operator norm and associated Lie algebras

This is a continuation of Crabb et al. (2021) [3] and Duncan and McGregor (2022) [4] in which we began an investigation of the real space H of Hermitian matrices in Mn(C) with respect to norms on Cn. Here, we introduce a family of Lie algebras derived from the classical real Lie algebras, with the aim of determining whether or not they are Lie isomorphic to ideals of some H.

Eigenproblem Basics and Algorithms

Some might say that the eigenproblem is one of the examples people discovered by looking at the sky and wondering. Even though it was formulated to explain the movement of the planets, today it has become the ansatz of solving many linear and nonlinear problems. Formulation in the terms of the eigenproblem is one of the key tools to solve complex problems, especially in the area of molecular geometry. However, the basic concept is difficult without proper preparation. A review paper covering basic concepts and algorithms is very useful. This review covers the basics of the topic. Definitions are provided for defective, Hermitian, Hessenberg, modal, singular, spectral, symmetric, skew-symmetric, skew-Hermitian, triangular, and Wishart matrices. Then, concepts of characteristic polynomial, eigendecomposition, eigenpair, eigenproblem, eigenspace, eigenvalue, and eigenvector are subsequently introduced. Faddeev–LeVerrier, von Mises, Gauss–Jordan, Pohlhausen, Lanczos–Arnoldi, Rayleigh–Ritz, Jacobi–Davidson, and Gauss–Seidel fundamental algorithms are given, while others (Francis–Kublanovskaya, Gram–Schmidt, Householder, Givens, Broyden–Fletcher–Goldfarb–Shanno, Davidon–Fletcher–Powell, and Saad–Schultz) are merely discussed. The eigenproblem has thus found its use in many topics. The applications discussed include solving Bessel’s, Helmholtz’s, Laplace’s, Legendre’s, Poisson’s, and Schrödinger’s equations. The algorithm extracting the first principal component is also provided.

Linearity of Cartan and Wasserstein means

The convex cone of positive definite Hermitian matrices has two important Riemannian geometries, where the parametrized weighted geometric (alternatively, Cartan) and Wasserstein means appear as the corresponding geodesics. The major problem with which this paper is concerned is linearity of Cartan and Wasserstein geodesics; when the Cartan (resp. Wasserstein) geodesic between two positive definite matrices A and B does lie in the space spanned by them and what the path in the plane to which it corresponds is. We settle the problem completely for the Cartan geometry and partially for the Wasserstein geometry. We show that their linearity problems for linearly independent A and B are equivalent to the solvability of the following equations for positive reals x and y with xy<14 respectively(xA+yB)−1=yA−1+xB−1,(AB)1/2+(BA)1/22=xA+yB.

Sectional nonassociativity of metrized algebras

The sectional nonassociativity of a metrized (not necessarily associative or unital) algebra is defined analogously to the sectional curvature of a pseudo-Riemannian metric, with the associator in place of the Levi-Civita covariant derivative. For commutative real algebras nonnegative sectional nonassociativity is usually called the Norton inequality, while a sharp upper bound on the sectional nonassociativity of the Jordan algebra of Hermitian matrices over a real Hurwitz algebra is closely related to the Böttcher–Wenzel-Chern-do Carmo-Kobayashi inequality. These and other basic examples are explained, and there are described some consequences of bounds on sectional nonassociativity for commutative algebras. A technical point of interest is that the results work over the octonions as well as the associative Hurwitz algebras.

A multi-GPU implementation of Real-Time Time-Dependent Auxiliary Density Functional Theory for the investigation of nanosystems irradiations

We report a new Multi-GPU (Graphical Processor Unit) implementation of real-time time-dependent Auxiliary Density Functional Theory (DFT) for simulations of attosecond electronic dynamics in molecular systems subjected to strong perturbations. Our code relies on the Kohn-Sham formalism of DFT and has been implemented in the deMon2k Fortran code. We expand single-particle wave functions (i.e molecular orbitals) as linear combinations of Gaussian-type-orbitals centered on atoms. The density matrix propagation is carried out on GPU while the Kohn-Sham potential is operated on CPUs (Central Processor Unit) with the help of variationally fitted densities. We propose a parallelization strategy using the MAGMA/CUDA libraries to calculate the exponential of dense Hermitian matrices entering the mathematical definition of the propagator, here using Taylor expansions. We report performance benchmarks on water droplets and on fullerenes (C50 to C540). They show a clear advantage of GPU over CPU (using the Scalapack library). The benchmarks also show the benefit of using more than one GPU for systems comprised of up to more than 10,000 basis functions. There, a speed-up of almost 40 between pure 40 CPU and four 4 GPU is obtained. Attosecond electron dynamics simulation in molecular systems comprised of several thousands of electrons becomes amenable to routine simulations in our code. We assess the accuracy of the GPU implementation considering various applications, namely, the calculation of extreme UV absorption spectra with non-Hermitian dynamics, the response of C180 to an electric perturbation, and finally the irradiation of a DNA/protein complex by a 0.4 MeV proton. The results demonstrate the robustness of the implementation. This work also paves the way for future even more efficient implementations. Program SummaryProgram title: deMon2k-RT-TDDFTCPC Library link to program files: https://doi.org/10.17632/whwc5zkst5.1Developer's repository link: DOI: 10.5281/zenodo.8301468Licensing provisions: CeCILL Free Software License Agreement v2.1Programming language: FortranExternal routines/libraries: MPI, BLAS/LAPACK, CUDA, MAGMASupplementary material: User manual, installation instructions, test cases. Nature of problem: Electron dynamics within molecular systems is amenable to simulation with Real-Time Time-Dependent Density Functional Theory (RT-TD-DFT). The methodology consists in propagating on the attosecond time scale the single particle electronic wave functions on the underlying Kohn-Sham potential [1,2]. In deMon2k-RT-TDDFT, the propagation is coupled to the Auxiliary DFT framework for fast evaluation of the potential [3]. A remaining computational bottleneck is the matrix exponential calculation of the Hermitian Hamiltonian matrix. This task is achieved in deMon2k via diagolanization or expansions formulas (Taylor, Chebychev or Baker-Campbell-Hausdorff). The operation is cumbersome on central-processor-unit machines for large matrices (e.g. > 5000 × 5000), which corresponds to molecular systems comprising around 2,000 electrons. Alternatives are needed to overcome this bottleneck.Solution method: We propose to carry out the matrix exponentiation on graphical-processor-units thanks to the MAGMA[4]/CUDA libraries. The code permits a drastic reduction of the computational cost, opening the attosecond dynamic simulations of large molecular systems comprised thousands of electrons with controlled accuracy. The method is tested here for Taylor expansions. Additional comments including restrictions and unusual features: The code described in this article opens the realization of RT-TD-ADFT simulations. The code is linked to the Master version 6.1.6 of deMon2k (http://www.demon-software.com), which must be pre-installed by the user.

Quantum algorithm for estimating largest eigenvalues

Scientific computation involving numerical methods relies heavily on the manipulation of large matrices, including solving linear equations and finding eigenvalues and eigenvectors. Quantum algorithms have been developed to advance these computational tasks, and some have been shown to provide substantial speedup, such as factoring a large integer and solving linear equations. In this work, we leverage the techniques used in the Harrow-Hassidim-Llyod (HHL) algorithm for linear systems, the classical power, and the Krylov subsapce method to devise a simple quantum algorithm for estimating the largest eigenvalues in magnitude of a Hermitian matrix. Our quantum algorithm offers significant speedup with respect to the size of a given matrix over classical algorithms that solve the same problem.

Universality of the cokernels of random 𝑝-adic Hermitian matrices

In this paper, we study the distribution of the cokernel of a general random Hermitian matrix over the ring of integers O \mathcal {O} of a quadratic extension K K of Q p \mathbb {Q}_p . For each positive integer n n , let X n X_n be a random n × n n \times n Hermitian matrix over O \mathcal {O} whose upper triangular entries are independent and their reductions are not too concentrated on certain values. We show that the distribution of the cokernel of X n X_n always converges to the same distribution which does not depend on the choices of X n X_n as n → ∞ n \rightarrow \infty and provide an explicit formula for the limiting distribution. This answers Open Problem 3.16 from the ICM 2022 lecture note of Wood [Probability theory for random groups arising in number theory, 2022] in the case of the ring of integers of a quadratic extension of Q p \mathbb {Q}_p .

Unitary canonical forms over Clifford algebras, and an observed unification of some real-matrix decompositions

We show that the spectral theorem – which we understand to be a statement that every self-adjoint matrix admits a certain type of canonical form under unitary similarity – admits analogues over other ∗ -algebras distinct from the complex numbers. If these ∗ -algebras contain nilpotents, then it is shown that there is a consistent way in which many classic matrix decompositions – such as the Singular Value Decomposition, the Takagi decomposition, the skew-Takagi decomposition, and the Jordan decomposition, among others – are immediate consequences of these. If producing the relevant canonical form of a self-adjoint matrix were a subroutine in some programming language, then the corresponding classic matrix decomposition would be a 1-line invocation with no additional steps. We also suggest that by employing operator overloading in a programming language, a numerical algorithm for computing a unitary diagonalization of a complex self-adjoint matrix would generalize immediately to solving problems like SVD or Takagi. While algebras without nilpotents (like the quaternions) allow for similar unifying behaviour, the classic matrix decompositions which they unify are never obtained as easily. In the process of doing this, we develop some spectral theory over Clifford algebras of the form C l p , q , 0 ( R ) and C l p , q , 1 ( R ) where the former is admittedly quite easy. We propose a broad conjecture about spectral theorems.