Symplectic Matrix Research Articles

On the moments of the moments of the characteristic polynomials of Haar distributed symplectic and orthogonal matrices

We establish formulae for the moments of the moments of the characteristic polynomials of random orthogonal and symplectic matrices in terms of certain lattice point count problems. This allows us to establish asymptotic formulae when the matrix-size tends to infinity in terms of the volumes of certain regions involving continuous Gelfand–Tsetlin patterns with constraints. The results we find differ from those in the unitary case considered previously.

On computing the symplectic LLT factorization

We analyze two algorithms for computing the symplectic factorization A = LLT of a given symmetric positive definite symplectic matrix A. The first algorithm W1 is an implementation of the HHT factorization from Dopico and Johnson (SIAM J. Matrix Anal. Appl. 31(2):650–673, 2009), see Theorem 5.2. The second one is a new algorithm W2 that uses both Cholesky and Reverse Cholesky decompositions of symmetric positive definite matrices. We present a comparison of these algorithms and illustrate their properties by numerical experiments in MATLAB. A particular emphasis is given on symplecticity properties of the computed matrices in floating-point arithmetic.

Random matrix theory and moments of moments of L-functions

In this paper, we give an analytic proof of the asymptotic behavior of the moments of moments of the characteristic polynomials of random symplectic and orthogonal matrices. We therefore obtain alternate, integral expressions for the leading order coefficients previously found by Assiotis, Bailey and Keating. We also discuss the conjectures of Bailey and Keating for the corresponding moments of moments of [Formula: see text]-functions with symplectic and orthogonal symmetry. Specifically, we show that these conjectures follow from the shifted moments conjecture of Conrey, Farmer, Keating, Rubinstein and Snaith.

On the surjectivity of the symplectic representation of the mapping class group

In this note, we study the symplectic representation of the mapping class group. In particular, we discuss the surjecivity of the representation restricted to certain mapping classes. It is well-known that the representation itself is surjective. In fact the representation is still surjective after restricting on pseudo-Anosov mapping classes. However, we show that the surjectivity does not hold when the representation is restricted on orientable pseudo-Anosovs, even after reducing its codomain to integer symplectic matrices with a bi-Perron leading eigenvalue. In order to prove the non-surjectivity, we explicitly construct an infinite family of symplectic matrices with a bi-Perron leading eigenvalue which cannot be obtained as the symplectic representation of an orientable pseudo-Anosov mapping class.

Wigner distribution associated with the symplectic coordinates transformation

The celebrated τ-Wigner distribution (τ-WD), a generalization of the ordinary Wigner distribution (WD), has attracted much attention in the literature for its wide use in time-frequency analysis. However, it fails to break through the time-frequency resolution limit of the WD. Due to more degrees of freedom, the matrix Wigner distribution (MWD), a generalization of the τ-WD and a special case of the A-Wigner distribution, seems to be an effective tool to tackle this challenge. In this study, we focus on a specific MWD, i.e., the so-called symplectic Wigner distribution (SWD), through the change of coordinates with the general symplectic matrix. We disclose an equivalence relation between the uncertainty product in SWD domains and that in the classical settings, from which we obtain the optimal symplectic matrix constraint condition under which the SWD achieves higher time-frequency resolution than the WD. We also provide the simulation example associated with linear frequency-modulated signals for verifying the established superresolution theory.

Analytical spectral design of mechanical metamaterials with inertia amplification

Functional metamaterials offering superior dynamic performances can be conceived by introducing local mechanisms of inertia amplification in the periodic microstructure of cellular composite media. The minimal physical realization of an inertially amplified metamaterial is represented by a one-dimensional crystal lattice, characterized by an intracellular pantograph mechanism. The microstructural dynamics of the periodic tetra-atomic cell is synthetically described by a low-dimension lagrangian model. The lagrangian coordinates account for the longitudinal motion of a pair of elastically coupled principal atoms, rigidly connected by the pantograph arms to a pair of transversely-oscillating secondary atoms, serving as inertial amplifiers. Within the range of small-oscillations, the free propagation of undamped harmonic waves is described by a linear difference equation, governed by a symplectic transfer matrix. First, the band structure of the complex-valued dispersion spectrum is determined analytically, by properly exploiting the formal (time-to-space) analogy with the Floquet Theory for the stability of non-autonomous dynamic systems (direct problem). Second, a spectral design problem is stated and solved by inverting analytically the functions expressing parametrically the boundaries separating attenuation (stop) and propagation (pass) bands in the frequency spectrum (inverse problem). Specifically, the inverse problem solution has the merit of providing simple formulas for parametric design. Such formulas explicitly and exactly identify the mechanical parameters of the inertially amplified metamaterial possessing a desired pass-stop-pass band structure, assigned according to functional design requirements. The existence and uniqueness of the solution in the admissible range of mechanical parameters is discussed. The discussion provides (i) mathematical demonstration for the existence of feasible design solutions or multi-solutions, (ii) complete definition of the physically realizable band structures in the frequency domain, and (iii) alternative criteria to design iso-band structured metamaterials within the admissible domain of mechanical parameters. The extra-customization possibilities offered by iso-band structured metamaterials are analyzed in terms of static and dynamic performances. As feasible limit cases of the analytical spectral design, extreme mechanical metamaterials working as phononic superfilters and/or superpropagators are realizable.

The Hamiltonian extended Krylov subspace method

An algorithm for constructing a $J$-orthogonal basis of the extended Krylov subspace$\mathcal{K}_{r,s}=\operatorname{range}\{u,Hu, H^2u,$ $ \ldots, $ $H^{2r-1}u, H^{-1}u, H^{-2}u, \ldots, H^{-2s}u\},$where $H \in \mathbb{R}^{2n \times 2n}$ is a large (and sparse) Hamiltonian matrix is derived (for $r = s+1$ or $r=s$). Surprisingly, this allows for short recurrences involving at most five previously generated basis vectors. Projecting $H$ onto the subspace $\mathcal{K}_{r,s}$ yields a small Hamiltonian matrix. The resulting HEKS algorithm may be used in order to approximate $f(H)u$ where $f$ is a function which maps the Hamiltonian matrix $H$ to, e.g., a (skew-)Hamiltonian or symplectic matrix. Numerical experiments illustrate that approximating $f(H)u$ with the HEKS algorithm is competitive for some functions compared to the use of other (structure-preserving) Krylov subspace methods.

Universal scaling limits of the symplectic elliptic Ginibre ensemble

We consider the eigenvalues of symplectic elliptic Ginibre matrices which are known to form a Pfaffian point process whose correlation kernel can be expressed in terms of the skew-orthogonal Hermite polynomials. We derive the scaling limits and the convergence rates of the correlation functions at the real bulk/edge of the spectrum, which in particular establishes the local universality at strong non-Hermiticity. Furthermore, we obtain the subleading corrections of the edge correlation kernels, which depend on the non-Hermiticity parameter contrary to the universal leading term. Our proofs are based on the asymptotic behavior of the complex elliptic Ginibre ensemble due to Lee and Riser as well as on a version of the Christoffel–Darboux identity, a differential equation satisfied by the skew-orthogonal polynomial kernel.

Rayleigh quotient and left eigenvalues of quaternionic matrices

We study the Rayleigh quotient of a Hermitian matrix with quaternionic coefficients and prove its main properties. As an application, we give some relationships between left and right eigenvalues of Hermitian and symplectic matrices.

Optimal unit triangular factorization of symplectic matrices

We prove that any symplectic matrix can be factored into no more than 5 unit triangular symplectic matrices, moreover, 5 is the optimal number. This result improves the existing triangular factorization of symplectic matrices which gives proof of 9 factors. We also show the corresponding improved conclusions for structured subsets of symplectic matrices. This factorization further provides an unconstrained optimization method on 2d-by-2d real symplectic group (a 2d2+d-dimensional Lie group) with 2d2+3d parameters.

Wigner analysis of operators. Part I: Pseudodifferential operators and wave fronts

We perform Wigner analysis of linear operators. Namely, the standard time-frequency representation Short-time Fourier Transform (STFT) is replaced by the A - Wigner distribution defined by W A ( f ) = μ ( A ) ( f ⊗ f ¯ ) , where A is a 4 d × 4 d symplectic matrix and μ ( A ) is an associate metaplectic operator. Basic examples are given by the so-called τ -Wigner distributions. Such representations provide a new characterization for modulation spaces when τ ∈ ( 0 , 1 ) . Furthermore, they can be efficiently employed in the study of the off-diagonal decay for pseudodifferential operators with symbols in the Sjöstrand class (in particular, in the Hörmander class S 0 , 0 0 ). The novelty relies on defining time-frequency representations via metaplectic operators, developing a conceptual framework and paving the way for a new understanding of quantization procedures. We deduce micro-local properties for pseudodifferential operators in terms of the Wigner wave front set. Finally, we compare the Wigner with the global Hörmander wave front set and identify the possible presence of a ghost region in the Wigner wave front. In the second part of the paper applications to Fourier integral operators and Schrödinger equations will be given.

Fluctuations of $$\beta $$-Jacobi product processes

We study Markov chains formed by squared singular values of products of truncated orthogonal, unitary, symplectic matrices (corresponding to the Dyson index $$\beta = 1,2,4$$ respectively) where time corresponds to the number of terms in the product. More generally, we consider the $$\beta $$ -Jacobi product process obtained by extrapolating to arbitrary $$\beta > 0$$ . For fixed time (i.e. number of factors is constant), we show that the global fluctuations are jointly Gaussian with explicit covariances. For time growing linearly with matrix size, we show convergence of moments after suitable rescaling. When $$\beta = 2$$ , our results imply that the right edge converges to a process which interpolates between the Airy point process and a deterministic configuration. This process connects a time-parametrized family of point processes appearing in the works of Akemann–Burda–Kieburg and Liu–Wang–Wang across time. In the arbitrary $$\beta > 0$$ case, our results show tightness of the particles near the right edge. The limiting moment formulas correspond to expressions for the Laplace transform of a conjectural $$\beta $$ -generalization of the interpolating process.

Sums of divisor functions and von Mangoldt convolutions in 𝔽 q [T] leading to symplectic distributions

Abstract In [J. P. Keating, B. Rodgers, E. Roditty-Gershon and Z. Rudnick, Sums of divisor functions in 𝔽 q ⁢ [ t ] \mathbb{F}_{q}[t] and matrix integrals, Math. Z. 288 2018, 1–2, 167–198], the authors established relationships of the mean-square of sums of the divisor function d k ⁢ ( f ) {d_{k}(f)} over short intervals and over arithmetic progressions for the function field 𝔽 q ⁢ [ T ] {\mathbb{F}_{q}[T]} to certain integrals over the ensemble of unitary matrices. We consider similar problems leading to distributions over the ensemble of symplectic matrices. We also consider analogous questions involving convolutions of the von Mangoldt function.

Oscillation Numbers for Continuous Lagrangian Paths and Maslov Index

In this paper we present the theory of oscillation numbers and dual oscillation numbers for continuous Lagrangian paths in \({\mathbb {R}}^{2n}\). Our main results include a connection of the oscillation numbers of the given Lagrangian path with the Lidskii angles of a special symplectic orthogonal matrix. We also present Sturmian type comparison and separation theorems for the difference of the oscillation numbers of two continuous Lagrangian paths. These results, as well as the definition of the oscillation number itself, are based on the comparative index theory (Elyseeva, 2009). The applications of these results are directed to the theory of Maslov index of two continuous Lagrangian paths. We derive a formula for the Maslov index via the Lidskii angles of a special symplectic orthogonal matrix, and hence we express the Maslov index as the oscillation number of a certain transformed Lagrangian path. The results and methods are based on a generalization of the recently introduced oscillation numbers and dual oscillation numbers for conjoined bases of linear Hamiltonian systems (Elyseeva, 2019 and 2020) and on the connection between the comparative index and Lidskii angles of symplectic matrices (Šepitka and Šimon Hilscher, 2021).

Complete symplectic quadrics and Kontsevich spaces of conics in Lagrangian Grassmannians

A wonderful compactification of an orbit under the action of a semi-simple and simply connected group is a smooth projective variety containing the orbit as a dense open subset, and where the added boundary divisor is simple normal crossing. We construct the wonderful compactification of the space of symmetric and symplectic matrices, and investigate its geometry. As an application, we describe the birational geometry of the Kontsevich spaces parametrizing conics in Lagrangian Grassmannians.

Application of Jordan canonical form and symplectic matrix in fractional differential models

Under consideration of this paper is the application of Jordan canonical form and symplectic matrix to two conformable fractional differential models. One is the new conformable fractional vector conduction equation which is reduced by using the Jordan canonical form of coefficient matrix and solved exactly, and the other is the new conformable fractional vector dynamical system with Hamilton matrix and symplectic matrix, which is derived by constructing the conformable fractional Euler-Lagrange equation and using fractional variational principle. It is shown that Jordan canonical form and symplectic matrix can be used to deal with some other conformable fractional differential systems in mathematical physics.

A novel symplectic relevance matrix machine method for intelligent fault diagnosis of roller bearing

In the fault classification of roller bearing, it is often encountered that the input sample is composed of feature matrix with rich structural information. As a new classifier, support matrix machine (SMM) makes full use of the structural information of input samples to establish prediction model. However, SMM lacks necessary probability information, and its sparsity and robustness are not clear. To resolve the above issues, a new matrix classifier - symplectic relevance matrix machine (SRMM) is proposed based on the probability framework and symplectic geometry theory. In SRMM, the model takes the original signal matrix as the input, and constructs the model elements with rich structural information. Then, symplectic geometry matrix is obtained by symplectic geometry similarity transformation, which makes SRMM robust. Meanwhile, the model elements are constructed by symplectic geometry, which can solve the difficulty of constructing recursive kernel function, reduce the complexity of the model, avoid the time consumption in the process of model parameter optimization. The experimental results of three roller bearing datasets show that SRMM has good classification performance in roller bearing fault diagnosis by comparing recognition rate, time, kappa, accuracy, recall rate and F1 and statistical test.

The relation between the symplectic group Sp(4,R) and its Lie algebra: Applications to polymer quantum mechanics

In this paper, we show the relation between $sp(4,\mathbb{R})$, the Lie algebra of the symplectic group, and the elements of $Sp(4,\mathbb{R})$. We use this result to obtain some special cases of symplectic matrices relevant to the study of squeezed states. In this regard, we provide some applications in quantum mechanics and analyze the squeezed polymer states obtained from the polymer representation of the symplectic group. Remarkably, the polymer's dispersions are the same as those obtained for the squeezed states in the usual representation.

Non-separable windowed linear canonical transform

The non-separable linear canonical transform (NSLCT) is a remarkable addition to the class of integral transforms which is based on generic 2n×2n free, symplectic matrix M with n(2n+1) degrees of freedom. However, the NSLCT is inadequate for localized analysis of non-transient signals, particularly chirp-like signals. In the present article, we introduce a novel integral transform coined as the non-separable windowed linear canonical transform (NSWLCT), which is endowed with higher degrees of freedom primarily meant for an efficient localized analysis of chirp signals. Firstly, we provide the time-frequency analysis of the proposed transform in the non-separable LCT domain. Secondly, we investigate the basic properties of the proposed transform including the orthogonality relation, inversion formula and the range theorem. Finally, we present examples of some well-known window functions in the non-separable LCT domain.

Williamson theorem in classical, quantum, and statistical physics

In this work, we present (and encourage the use of) the Williamson theorem and its consequences in several contexts in physics. We demonstrate this theorem using only basic concepts of linear algebra and symplectic matrices. As an immediate application in the context of small oscillations, we show that applying this theorem reveals the normal-mode coordinates and frequencies of the system in the Hamiltonian scenario. A modest introduction of the symplectic formalism in quantum mechanics is presented, using the theorem to study quantum normal modes and canonical distributions of thermodynamically stable systems described by quadratic Hamiltonians. As a last example, a more advanced topic concerning uncertainty relations is developed to show once more its utility in a distinct and modern perspective.