Symmetric Positive Semidefinite Matrix Research Articles

Real factorization of positive semidefinite matrix polynomials

Suppose Q(x) is a real n×n regular symmetric positive semidefinite matrix polynomial. Then it can be factored asQ(x)=G(x)TG(x), where G(x) is a real n×n matrix polynomial with degree half that of Q(x) if and only if det⁡(Q(x)) is the square of a nonzero real polynomial. We provide a constructive proof of this fact, rooted in finding a skew-symmetric solution to a modified algebraic Riccati equationXSX−XR+RTX+P=0, where P,R,S are real n×n matrices with P and S real symmetric. In addition, we provide a detailed algorithm for computing the factorization.

Computation of the von Neumann entropy of large matrices via trace estimators and rational Krylov methods

We consider the problem of approximating the von Neumann entropy of a large, sparse, symmetric positive semidefinite matrix A, defined as {{,textrm{tr},}}(f(A)) where f(x)=-xlog x. After establishing some useful properties of this matrix function, we consider the use of both polynomial and rational Krylov subspace algorithms within two types of approximations methods, namely, randomized trace estimators and probing techniques based on graph colorings. We develop error bounds and heuristics which are employed in the implementation of the algorithms. Numerical experiments on density matrices of different types of networks illustrate the performance of the methods.

The Moore-Penrose inverse of the distance matrix of a helm graph

In this paper, we give necessary and sufficient conditions for a real symmetric matrix and, in particular, for the distance matrix $D(H_n)$ of a helm graph $H_n$ to have their Moore-Penrose inverses as the sum of a symmetric Laplacian-like matrix and a rank-one matrix. As a consequence, we present a short proof of the inverse formula, given by Goel (Linear Algebra Appl. 621:86-104, 2021), for $D(H_n)$ when $n$ is even. Further, we derive a formula for the Moore-Penrose inverse of singular $D(H_n)$ that is analogous to the formula for $D(H_n)^{-1}$. Precisely, if $n$ is odd, we find a symmetric positive semi-definite Laplacian-like matrix $L$ of order $2n-1$ and a vector $\mathbf{w}\in \mathbb{R}^{2n-1}$ such that\begin{eqnarray*}D(H_n)^{\dagger} = -\frac{1}{2}L +\frac{4}{3(n-1)}\mathbf{w}\mathbf{w^{\prime}},\end{eqnarray*}where the rank of $L$ is $2n-3$. We also investigate the inertia of $D(H_n)$.

Au26-35: A Special Geometrical Structure of Au33 (D2) Cluster with Highly Occupied - 14 Pairs of Double-State Degeneracy

In this article, an interesting phenomenon has described the geometries and vibrational frequency of the stable Aun clusters with n = 26 to 35. We have found nine out of ten clusters are having the very same C1 symmetry, except the cluster n = 33 (D2). The finite-differentiation method has been implemented within the density-functional tight-binding (DFTB) approach. The effects of the range of interatomic forces were calculated and the desired set of system eigenfrequencies (3N-6) are obtained by diagonalization of the symmetric positive semidefinite Hessian matrix. Mainly, we have observed the vibrational spectra and the range comes in between 2.04 and 347.32 cm−1 at ground state, ∆E = 0. Most significantly, all the clusters have revealed double-state degeneracy. The vibrational spectrum is strongly dependent upon the size, shape, and structure, at the same time, the stretching and the bending mode of the atoms with respect to the bond length plays a major role. We have compared some of our results, which have an excellent agreement, with the less availability of the experimental and the theoretical predictions.

Improved Variants of the Hutch++ Algorithm for Trace Estimation

This paper is concerned with two improved variants of the Hutch++ algorithm for estimating the trace of a square matrix, implicitly given through matrix-vector products. Hutch++ combines randomized low-rank approximation in a first phase with stochastic trace estimation in a second phase. In turn, Hutch++ only requires $O\left(\varepsilon^{-1}\right)$ matrix-vector products to approximate the trace within a relative error $\varepsilon$ with high probability. This compares favorably with the $O\left(\varepsilon^{-2}\right)$ matrix-vector products needed when using stochastic trace estimation alone. In Hutch++, the number of matrix-vector products is fixed a priori and distributed in a prescribed fashion among the two phases. In this work, we derive an adaptive variant of Hutch++, which outputs an estimate of the trace that is within some prescribed error tolerance with a controllable failure probability, while splitting the matrix-vector products in a near-optimal way among the two phases. For the special case of symmetric positive semi-definite matrix, we present another variant of Hutch++, called Nystr\om++, which utilizes the so called Nystr\om approximation and requires only one pass over the matrix, as compared to two passes with Hutch++. We extend the analysis of Hutch++ to Nystr\om++. Numerical experiments demonstrate the effectiveness of our two new algorithms.

Damping Optimization of Linear Vibrational Systems with a Singular Mass Matrix

We present two novel results for small damped oscillations described by the vector differential equation Mx¨+Cx˙+Kx=0, where the mass matrix M can be singular, but standard deflation techniques cannot be applied. The first result is a novel formula for the solution X of the Lyapunov equation ATX+XA=−I, where A=A(v) is obtained from M,C(v)∈Rn×n, and K∈Rn×n, which are the so-called mass, damping, and stiffness matrices, respectively, and rank(M)=n−1. Here, C(v) is positive semidefinite with rank(C(v))=1. Using the obtained formula, we propose a very efficient way to compute the optimal damping matrix. The second result was obtained for a different structure, where we assume that dim(N(M))≥1 and internal damping exists (usually a small percentage of the critical damping). For this structure, we introduce a novel linearization, i.e., a novel construction of the matrix A in the Lyapunov equation ATX+XA=−I, and a novel optimization process. The proposed optimization process computes the optimal damping C(v) that minimizes a function v↦trace(ZX) (where Z is a chosen symmetric positive semidefinite matrix) using the approximation function g(v)=cv+av+bv, for the trace function f(v)≐trace(ZX(v)). Both results are illustrated with several corresponding numerical examples.

On sums of squares of k-nomials

In 2005, Boman et al. introduced the concept of factor width for a real symmetric positive semidefinite matrix . This is the smallest positive integer k for which the matrix A can be written as A = V V T with each column of V containing at most k non-zeros. The cones of matrices of bounded factor width give a hierarchy of inner approximations to the PSD cone. In the polynomial optimization context, a Gram matrix of a polynomial having factor width k corresponds to the polynomial being a sum of squares of polynomials of support at most k . Recently, Ahmadi and Majumdar [1] , explored this connection for case k = 2 and proposed to relax the reliance on polynomials that are sums of squares in semidefinite programming to polynomials that are sums of binomial squares In this paper, we prove some results on the geometry of the cones of matrices with bounded factor widths and their duals, and use them to derive new results on the limitations of certificates of nonnegativity of quadratic forms by sums of k -nomial squares using standard multipliers. In particular we show that they never help for symmetric quadratics, for any quadratic if k = 2 , and any quaternary quadratic if k = 3 . Furthermore we give some evidence that those are a complete list of such cases.

Expansion in matrix-weighted graphs

A matrix-weighted graph is an undirected graph with a k×k symmetric positive semidefinite matrix assigned to each edge. Such graphs admit natural generalizations of the Laplacian and adjacency matrices, leading to a generalized notion of expansion. Extensions of some theorems about expansion hold for matrix-weighted graphs—in particular, an analogue of the expander mixing lemma and one half of a Cheeger-type inequality. These results lead to a definition of a matrix-weighted expander graph, and suggest the tantalizing possibility of families of matrix-weighted graphs with better-than-Ramanujan expansion.

Lump and rational solutions for weakly coupled generalized Kadomtsev–Petviashvili equation

Through the [Formula: see text]-KP hierarchy, we present a new (3+1)-dimensional equation called weakly coupled generalized Kadomtsev–Petviashvili (wc-gKP) equation. Based on Hirota bilinear differential equations, we get rational solutions to wc-gKP equation, and further we obtain lump solutions by searching for a symmetric positive semi-definite matrix. We do some numerical analysis on the trajectory of rational solutions and fit the trajectory equation of wave crest. Some graphics are illustrated to describe the properties of rational solutions and lump solutions. The method used in this paper to get lump solutions by constructing a symmetric positive semi-definite matrix can be applied to other integrable equations as well. The results expand the understanding of lump and rational solutions in soliton theory.

Some algorithms for maximum volume and cross approximation of symmetric semidefinite matrices

Various applications in numerical linear algebra and computer science are related to selecting the rtimes r submatrix of maximum volume contained in a given matrix Ain mathbb R^{ntimes n}. We propose a new greedy algorithm of cost mathcal O(n), for the case A symmetric positive semidefinite (SPSD) and we discuss its extension to related optimization problems such as the maximum ratio of volumes. In the second part of the paper we prove that any SPSD matrix admits a cross approximation built on a principal submatrix whose approximation error is bounded by (r+1) times the error of the best rank r approximation in the nuclear norm. In the spirit of recent work by Cortinovis and Kressner we derive some deterministic algorithms, which are capable to retrieve a quasi optimal cross approximation with cost mathcal O(n^3).

Some log and weak majorization inequalities in Euclidean Jordan algebras

Motivated by Horn's log-majorization (singular value) inequality and the related weak-majorization inequality for square complex matrices, we consider their Hermitian analogs for positive semidefinite matrices and for general (Hermitian) matrices, where denotes the Jordan product of A and B and denotes the componentwise product in . In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form for and for all a and b, where and denote, respectively, the quadratic representation and the eigenvalue vector of an element u. We also describe inequalities of the form , where A is a real symmetric positive semidefinite matrix and is the Schur product of A and b. In the form of an application, we prove the generalized Hölder type inequality , where denotes the spectral p-norm of x and with . We also give precise values of the norms of the Lyapunov transformation and relative to two spectral p-norms.

Weakly Coupled B-Type Kadomtsev-Petviashvili Equation: Lump and Rational Solutions

Through the method of Z N -KP hierarchy, we propose a new ( 3 + 1 )-dimensional weakly coupled B-KP equation. Based on the bilinear form, we obtain the lump and rational solutions to the dimensionally reduced cases by constructing a symmetric positive semidefinite matrix. Then, we do numerical analysis on the rational solutions and fit the trajectory equation of the crest. Furthermore, we verify the accuracy of the trajectory equation by numerical analysis. This method of solving the lump and rational solutions can also be applied to other nonlinear evolution equations.

Two Relaxation Methods for Rank Minimization Problems

The problem of minimizing the rank of a symmetric positive semidefinite matrix subject to constraints can be lifted to give an equivalent semidefinite program with complementarity constraints. The formulation requires two positive semidefinite matrices to be complementary. This is a continuous and nonconvex reformulation of the rank minimization problem. We develop two relaxations and show that constraint qualification holds at any stationary point of either relaxation of the rank minimization problem, and we explore the structure of the local minimizers.

Novel Geometrically Defined Mass Matrices for Tetrahedral Meshes

This contribution introduces a novel method for constructing mass matrices for general tetrahedral elements. By considering the construction of the reluctivity mass matrix as an example, a new derivation of a recipe to geometrically construct a symmetric positive semi-definite and consistent mass matrix is provided. It is shown why such a matrix can be used inside formulations where the mass matrix is right multiplied by the appropriate incidence matrix. Then, a second recipe is introduced to produce a consistent and symmetric positive definite matrix. The proposed method is based on an original algebraic technique that differs fundamentally from what is available in the literature. The proposed recipe opens also the possibility to improve the conditioning of the resulting elementary mass matrices.

Auto-Gauging of Vector Potential by Parallel Sparse Direct Solvers—Numerical Observations

When using magnetic vector potential (MVP)-based formulations for magnetostatic or eddy-current problems, either gauge conditions specifying the divergence of the MVP or tree gauging by eliminating redundant degrees of freedom of the MVP is usually imposed to ensure uniqueness of solutions. Explicit gauging of the MVP is not always necessary since classical iterative solvers can automatically and implicitly fix the gauge as long as the right-hand side vectors are consistent. Besides iterative solvers, implicit gauging is also observed when using state-of-the-art parallel sparse direct solvers (PSDSs), thanks to the built-in functions of handling null-spaces of either real symmetric positive semi-definite matrix systems or those complex symmetric systems from eddy-current problems. Both static and eddy-current examples are solved by PSDS to demonstrate results of local physical quantities or global quantities such as magnetic energy or joule losses. High-order edge/nodal elements are also considered in our numerical examples and it is observed that PSDS can also easily and correctly handle the delicate discrete null spaces.

Improved GNN Models for Constant Matrix Inversion

It was shown that the multiplication of the left hand side of the classical Zhang neural network design rule by an appropriate positive definite matrix generates a new neural design with improved convergence rate. Our intention is to apply similar principle on the standard gradient neural network (GNN) model. To that goal, we discover that some of proposed models can be considered as the multiplication of the right hand side of the GNN model by a symmetric positive-semidefinite matrix. As a final result, we propose appropriate general pattern to define various improvements of the standard GNN design for online real-time matrix inversion in time invariant case. The leading idea in generating improved models is initiated after a combination of two GNN patterns. Improved GNN (IGNN) design shows global exponential convergence with an improved convergence rate with respect to the convergence rate of the original GNN pattern. The acceleration in the convergence rate is defined by the smallest eigenvalue of appropriate positive semidefinite matrices. IGNN models are not only generalizations of original GNN models but they comprise so far defined improvements of the standard GNN design.

Efficient Energy-preserving Methods for General Nonlinear Oscillatory Hamiltonian System

In this paper, we propose and analyze two kinds of novel and symmetric energy-preserving formulae for the nonlinear oscillatory Hamiltonian system of second-order differential equations Aq″ (t)+ Bq(t) = f(q(t)), where A ∈ ℝm×m is a symmetric positive definite matrix, B ∈ ℝm×m is a symmetric positive semi-definite matrix that implicitly contains the main frequencies of the problem and f(q) = −∇qV (q) for a real-valued function V (q). The energy-preserving formulae can exactly preserve the Hamiltonian $$H(q', q) = \frac{1}{2}q'^{\rm{T}} Aq' + \frac{1}{2}q^{\rm{T}} Bq + V(q)$$ . We analyze the properties of energy-preserving and convergence of the derived energy-preserving formula and obtain new efficient energy-preserving integrators for practical computation. Numerical experiments are carried out to show the efficiency of the new methods by the nonlinear Hamiltonian systems.

Existence and concentration of solution for a class of fractional Hamiltonian systems with subquadratic potential

In this article, we consider the following fractional Hamiltonian systems: $$\begin{aligned} {_{t}}D_{\infty }^{\alpha }({_{-\infty }}D_{t}^{\alpha }u) + \lambda L(t)u = \nabla W(t, u), \;\;t\in \mathbb {R}, \end{aligned}$$ where $$\alpha \in (1/2, 1)$$ , $$\lambda >0$$ is a parameter, $$L\in C(\mathbb {R}, \mathbb {R}^{n\times n})$$ and $$W \in C^{1}(\mathbb {R} \times \mathbb {R}^n, \mathbb {R})$$ . Unlike most other papers on this problem, we require that L(t) is a positive semi-definite symmetric matrix for all $$t\in \mathbb {R}$$ , that is, $$L(t) \equiv 0$$ is allowed to occur in some finite interval $$\mathbb {I}$$ of $$\mathbb {R}$$ . Under some mild assumptions on W, we establish the existence of nontrivial weak solution, which vanish on $$\mathbb {R} \setminus \mathbb {I}$$ as $$\lambda \rightarrow \infty ,$$ and converge to $$\tilde{u}$$ in $$H^{\alpha }(\mathbb {R})$$ ; here $$\tilde{u} \in E_{0}^{\alpha }$$ is nontrivial weak solution of the Dirichlet BVP for fractional Hamiltonian systems on the finite interval $$\mathbb {I}$$ . Furthermore, we give the multiplicity results for the above fractional Hamiltonian systems.

Efficient computation of matrix power-vector products: Application for space-fractional diffusion problems

A novel algorithm is proposed for computing matrix–vector products Aαv, where A is a symmetric positive semidefinite sparse matrix and α>0. The method can be applied for the efficient implementation of the matrix transformation method to solve space-fractional diffusion problems. The performance of the new algorithm is studied in a comparison with the conventional MATLAB subroutines to compute matrix powers.

A penalty method for rank minimization problems in symmetric matrices

The problem of minimizing the rank of a symmetric positive semidefinite matrix subject to constraints can be cast equivalently as a semidefinite program with complementarity constraints (SDCMPCC). The formulation requires two positive semidefinite matrices to be complementary. This is a continuous and nonconvex reformulation of the rank minimization problem. We investigate calmness of locally optimal solutions to the SDCMPCC formulation and hence show that any locally optimal solution is a KKT point. We develop a penalty formulation of the problem. We present calmness results for locally optimal solutions to the penalty formulation. We also develop a proximal alternating linearized minimization (PALM) scheme for the penalty formulation, and investigate the incorporation of a momentum term into the algorithm. Computational results are presented.