skew-Hermitian Research Articles

Cayley transform for Toeplitz and dual matrices

Let an n×n complex matrix A be such that I+A is invertible. The Cayley transform of A, denoted by C(A), is defined asC(A)=(I+A)−1(I−A). Fallat and Tsatsomeros (2002) [5] and Mondal et al. (2024) [15] studied the Cayley transform of a matrix A in the context of P-matrices, H-matrices, M-matrices, totally positive matrices, positive definite matrices, almost skew-Hermitian matrices, and semipositive matrices. In this paper, the investigation of the Cayley transform is continued for Toeplitz matrices, circulant matrices, unipotent matrices, and dual matrices. An expression of the Cayley transform for dual matrices is established. It is shown that the Cayley transform of a dual symmetric matrix is always a dual symmetric matrix. The Cayley transform of a dual skew-symmetric matrix is discussed. The results are illustrated with examples.

The c-numerical range of a quaternion skew-Hermitian matrix is convex

We show that the c-numerical range of a non-scalar skew-Hermitian quaternion matrix is convex. In fact, included in our result is that the c-numerical range of a skew-Hermitian matrix is a rotation invariant subset of the quaternions with zero real parts.

Some New Algebraic Method Developments in the Characterization of Matrix Equalities

Algebraic expressions and equalities can be constructed arbitrarily in a given algebraic framework according to the operational rules provided, and thus it is a prominent and necessary task in mathematics and applications to construct, classify, and characterize various simple general algebraic expressions and equalities. As an update to this prominent topic in matrix algebra, this article reviews and improves upon the well-known block matrix methodology and matrix rank methodology in the construction and characterization of matrix equalities. We present a collection of fundamental and useful formulas for calculating the ranks of a wide range of block matrices and then derive from these rank formulas various valuable consequences. In particular, we present several groups of equivalent conditions in the characterizations of the Hermitian matrix, the skew-Hermitian matrix, the normal matrix, etc.

A systematic methodology for port-Hamiltonian modeling of multidimensional flexible linear mechanical systems

This article introduces a novel systematic methodology for modeling a class of multidimensional linear mechanical systems that directly allows to obtain their infinite-dimensional port-Hamiltonian representation. While the approach is tailored to systems governed by specific kinematic assumptions, it encompasses a wide range of models found in current literature, including ℓ-dimensional elasticity models (where ℓ = 1, 2, 3), vibrating strings, torsion in circular bars, classical beam and plate models, among others. The methodology involves formulating the displacement field using primary generalized coordinates via a linear algebraic relation. The non-zero components of the strain tensor are then calculated and expressed using secondary generalized coordinates, enabling the characterization of the skew-adjoint differential operator associated with the port-Hamiltonian representation. By applying Hamilton's principle and employing a specially developed integration by parts formula for the considered class of differential operators, the port-Hamiltonian model is directly obtained, along with the definition of boundary inputs and outputs. To illustrate the methodology, the plate modeling process based on Reddy's third-order shear deformation theory is presented as an example. To the best of our knowledge, this is the first time that a port-Hamiltonian representation of this system is presented in the literature.

Consistent spectral approximation of Koopman operators using resolvent compactification

Koopman operators and transfer operators represent dynamical systems through their induced linear action on vector spaces of observables, enabling the use of operator-theoretic techniques to analyze nonlinear dynamics in state space. The extraction of approximate Koopman or transfer operator eigenfunctions (and the associated eigenvalues) from an unknown system is nontrivial, particularly if the system has mixed or continuous spectrum. In this paper, we describe a spectrally accurate approach to approximate the Koopman operator on L 2 for measure-preserving, continuous-time systems via a ‘compactification’ of the resolvent of the generator. This approach employs kernel integral operators to approximate the skew-adjoint Koopman generator by a family of skew-adjoint operators with compact resolvent, whose spectral measures converge in a suitable asymptotic limit, and whose eigenfunctions are approximately periodic. Moreover, we develop a data-driven formulation of our approach, utilizing data sampled on dynamical trajectories and associated dictionaries of kernel eigenfunctions for operator approximation. The data-driven scheme is shown to converge in the limit of large training data under natural assumptions on the dynamical system and observation modality. We explore applications of this technique to dynamical systems on tori with pure point spectra and the Lorenz 63 system as an example with mixing dynamics.

Qubit Count Reduction by Orthogonally Constrained Orbital Optimization for Variational Quantum Excited-State Solvers.

We propose a state-averaged orbital optimization scheme for improving the accuracy of excited states of the electronic structure Hamiltonian for use on near-term quantum computers. Instead of parameterizing the orbital rotation operator in the conventional fashion as an exponential of an antihermitian matrix, we parameterize the orbital rotation as a general partial unitary matrix. Whereas conventional orbital optimization methods minimize the state-averaged energy using successive Newton steps of the second-order Taylor expansion of the energy, the method presented here optimizes the state-averaged energy using an orthogonally constrained gradient projection method that does not require any expansion approximations. Through extensive benchmarking of the method on various small molecular systems, we find that the method is capable of producing more accurate results than fixed basis FCI while simultaneously using fewer qubits. In particular, we show that for H2, the method is capable of matching the accuracy of FCI in the cc-pVTZ basis (56 qubits) while only using 14 qubits.

On Fourier expansions for systems of ordinary differential equations with distributional coefficients

We study the spectral theory for the first-order system Ju′+qu=wf of differential equations on the real interval (a,b) where J is a constant, invertible, skew-hermitian matrix and q and w are matrices whose entries are distributions of order 0 with q hermitian and w non-negative. Specifically, we construct a generalized Weyl-Titchmarsh m-function with corresponding spectral measure τ and a generalized Fourier transform after imposing certain conditions on J, q, and w. Different conditions are motivated and studied in the later sections. A Fatou-type identity needed for our result is recorded in the appendix.

Jet space extensions of infinite-dimensional Hamiltonian systems

We analyze infinite-dimensional Hamiltonian systems corresponding to partial Differential equations on one-dimensional spatial domains formulated with formally skew-adjoint Hamiltonian operators and non-quadratic Hamiltonian density. In various applications, the Hamiltonian density can depend on spatial derivatives of the state such that these systems can not straightforwardly be formulated as boundary port-Hamiltonian system using a Stokes-Dirac structure. In this work, we show that any Hamiltonian system of the above class can be reformulated as a Hamiltonian system on the jet space, in which the Hamiltonian density only depends on the extended state variable itself and not on its derivatives. Consequently, well-known geometric formulations with Stokes-Dirac structures are applicable. Additionally, we provide a similar result for dissipative systems. We illustrate the developed theory by means of the the Boussinesq equation, the dynamics of an elastic rod and the Allen-Cahn equation.

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 .

The Cayley transform of prevalent matrix classes

The Cayley transform of a matrix A, C(A)=(I+A)−1(I−A), has been investigated in the contexts of skew-symmetry, eigenvalue mapping and as a factorization technique for several matrix positivity classes. Here, we extend the discussion to Cayley transforms of almost skew-Hermitian matrices, positive definite matrices, semipositive matrices, as well as subclasses of the M-matrices, H-matrices and their inverses.

Orientation flow for skew-adjoint Fredholm operators with odd-dimensional kernel

The orientation flow of paths of real skew-adjoint Fredholm operators with invertible endpoints was studied by Carey, Phillips and Schulz-Baldes. For paths of real skew-adjoint Fredholm operators with odd-dimensional kernel the orientation flow is defined with respect to a real one-dimensional reference projection. It is homotopy invariant and fulfills a concatenation property. When applied to closed paths it is independent of the reference projection and provides an isomorphism of the fundamental group of the space of real skew-adjoint Fredholm operators with odd-dimensional kernel to Z2. As an example the orientation flow of the magnetic flux inserted in a half-sided Kitaev chain is studied.

Weakly invariant norms: Geometry of spheres in the space of skew-Hermitian matrices

Let N be a weakly unitarily invariant norm (i.e. invariant for the coadjoint action of the unitary group) in the space of skew-Hermitian matrices un(C). In this paper we study the geometry of the unit sphere of such a norm, and we show how its geometric properties are encoded by the majorization properties of the eigenvalues of the matrices. We give a detailed characterization of norming functionals of elements for a given norm, and we then prove a sharp criterion for the commutator [X,[X,V]] to be in the hyperplane that supports V in the unit sphere. We show that the adjoint action V↦V+[X,V] of un(C) on itself pushes vectors away from the unit sphere. As an application of the previous results, for a strictly convex norm, we prove that the norm is preserved by this last action if and only if X commutes with V. We give a more detailed description in the case of any weakly Ad-invariant norm.

The energy method for high-order invariants in shallow water wave equations

Third-order dispersive evolution equations are widely adopted to model one-dimensional long waves and have extensive applications in fluid mechanics, plasma physics and nonlinear optics. The typical representatives are the KdV equation, the Camassa–Holm equation and the Degasperis–Procesi equation. They share many common features such as complete integrability, Lax pairs and bi-Hamiltonian structure. In this paper we revisit high-order invariants for these three types of shallow water wave equations by the energy method in combination of a skew-adjoint operator (1−∂xx)−1. Several applications to seek high-order invariants of the Benjamin–Bona–Mahony equation, the regularized long-wave equation and the Rosenau equation are also presented.

Gradient Flows, Adjoint Orbits, and the Topology of Totally Nonnegative Flag Varieties

One can view a partial flag variety in \({\mathbb {C}}^n\) as an adjoint orbit \({\mathcal {O}}_{\lambda }\) inside the Lie algebra of \(n\times n\) skew-Hermitian matrices. We use the orbit context to study the totally nonnegative part of a partial flag variety from an algebraic, geometric, and dynamical perspective. The paper has three main parts: (1) We introduce the totally nonnegative part of \({\mathcal {O}}_{\lambda }\), and describe it explicitly in several cases. We define a twist map on it, which generalizes (in type A) a map of Bloch, Flaschka, and Ratiu (Duke Math. J. 61(1): 41–65, 1990) on an isospectral manifold of Jacobi matrices. (2) We study gradient flows on \({\mathcal {O}}_{\lambda }\) which preserve positivity, working in three natural Riemannian metrics. In the Kähler metric, positivity is preserved in many cases of interest, extending results of Galashin, Karp, and Lam (Adv. Math. 397: Paper No. 108123, 1–23, 2022; Adv. Math. 351: 614–620, 2019). In the normal metric, positivity is essentially never preserved on a generic orbit. In the induced metric, whether positivity is preserved appears to depends on the spacing of the eigenvalues defining the orbit. (3) We present two applications. First, we discuss the topology of totally nonnegative flag varieties and amplituhedra. Galashin, Karp, and Lam (2022, 2019) showed that the former are homeomorphic to closed balls, and we interpret their argument in the orbit framework. We also show that a new family of amplituhedra, which we call twisted Vandermonde amplituhedra, are homeomorphic to closed balls. Second, we discuss the symmetric Toda flow on \({\mathcal {O}}_{\lambda }\). We show that it preserves positivity, and that on the totally nonnegative part, it is a gradient flow in the Kähler metric up to applying the twist map. This extends a result of Bloch, Flaschka, and Ratiu (1990).

Closed-form formula for a classical system of matrix equations

Keeping in view the latest development of anti-Hermitian matrix in mind, we construct some closed form formula for a classical system of matrix equations having anti-Hermitian nature in this paper. We give the necessary and sufficient conditions for the existence of its solution by applying the properties of matrix rank. The general solution to this system is expressed by closed formula based on generalized inverses of given matrices. The novelty of the proposed results is not only obtaining a formal representation of the solution in terms of generalized inverses but the construction of an algorithm to find its explicit expression as well. To conduct an algorithm and numerical example, it is used the determinantal representations of the Moore Penrose inverse previously obtained by one of the authors.

Modulus-based matrix splitting iteration methods with new splitting scheme for horizontal implicit complementarity problems

In this paper, we propose the horizontal implicit complementarity problems, which cover many complementarity problems that have wide applications. After reformulating the horizontal implicit complementarity problem as an implicit fixed-point equation, the modulus-based matrix splitting iteration methods are applied to solve the problems. Besides, this paper introduces a new kind of matrix splitting, called modulus-based Hermitian and skew-Hermitian matrix relaxation splitting. The corresponding modulus-based matrix splitting iteration method is shown superior to the nonsmooth Newton's method and the existing matrix splitting schemes by some numerical experiments. The convergence analysis is given and validates that our methods are more practical in applications.

Green’s Functions for First-Order Systems of Ordinary Differential Equations without the Unique Continuation Property

This paper is a contribution to the spectral theory associated with the differential equation $Ju'+qu=wf$ on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. Under these hypotheses it may not be possible to uniquely continue a solution from one point to another, thus blunting the standard tools of spectral theory. Despite this fact we are able to describe symmetric restrictions of the maximal relation associated with $Ju'+qu=wf$ and show the existence of Green's functions for self-adjoint relations even if unique continuation of solutions fails.

Real dimension of the Lie algebra of S-skew-Hermitian matrices

Let $M_n(\mathbb{C})$ be the set of $n\times n$ matrices over the complex numbers. Let $S \in M_n(\mathbb{C})$. A matrix $A\in M_n(\mathbb{C})$ is said to be $S$-skew-Hermitian if $SA^*=-AS$ where $A^*$ is the conjugate transpose of $A$. The set $\mathfrak{u}_S$ of all $S$-skew-Hermitian matrices is a Lie algebra. In this paper, we give a real dimension formula for $\mathfrak{u}_S$ using the Jordan block decomposition of the cosquare $S(S^*)^{-1}$ of $S$ when $S$ is nonsingular.

An efficient algorithm to compute the exponential of skew-Hermitian matrices for the time integration of the Schrödinger equation

We present a practical algorithm to approximate the exponential of skew-Hermitian matrices up to round-off error based on an efficient computation of Chebyshev polynomials of matrices and the corresponding error analysis. It is based on Chebyshev polynomials of degrees 2, 4, 8, 12 and 18 which are computed with only 1, 2, 3, 4 and 5 matrix–matrix products, respectively. For problems of the form exp(−iA), with A a real and symmetric matrix, an improved version is presented that computes the sine and cosine of A with a reduced computational cost. The theoretical analysis, supported by numerical experiments, indicates that the new methods are more efficient than schemes based on rational Padé approximants and Taylor polynomials for all tolerances and time interval lengths. The new procedure is particularly recommended to be used in conjunction with exponential integrators for the numerical time integration of the Schrödinger equation.

A study of defect-based error estimates for the Krylov approximation of \u03c6-functions

Prior recent work, devoted to the study of polynomial Krylov techniques for the approximation of the action of the matrix exponential etAv, is extended to the case of associated φ-functions (which occur within the class of exponential integrators). In particular, a posteriori error bounds and estimates, based on the notion of the defect (residual) of the Krylov approximation are considered. Computable error bounds and estimates are discussed and analyzed. This includes a new error bound which favorably compares to existing error bounds in specific cases. The accuracy of various error bounds is characterized in relation to corresponding Ritz values of A. Ritz values yield properties of the spectrum of A (specific properties are known a priori, e.g., for Hermitian or skew-Hermitian matrices) in relation to the actual starting vector v and can be computed. This gives theoretical results together with criteria to quantify the achieved accuracy on the fly. For other existing error estimates, the reliability and performance are studied by similar techniques. Effects of finite precision (floating point arithmetic) are also taken into account.