Space Of Real Symmetric Matrices Research Articles

On an analogue of a property of singular M-matrices for the Lyapunov and Stein operators

A well-known result for a singular irreducible M-matrix A is that the only nonnegative vector that belongs to the range space of A is the zero vector. In this paper, we prove an analogue of this result for the Lyapunov and Stein transformations, which act on the inner product space of real symmetric matrices.

The middle-scale asymptotics of Wishart matrices

We study the behavior of a real $p$-dimensional Wishart random matrix with $n$ degrees of freedom when $n,p\rightarrow\infty$ but $p/n\rightarrow0$. We establish the existence of phase transitions when $p$ grows at the order $n^{(K+1)/(K+3)}$ for every $K\in\mathbb{N}$, and derive expressions for approximating densities between every two phase transitions. To do this, we make use of a novel tool we call the $\mathcal{F}$-conjugate of an absolutely continuous distribution, which is obtained from the Fourier transform of the square root of its density. In the case of the normalized Wishart distribution, this represents an extension of the $t$-distribution to the space of real symmetric matrices.

The Extended Matrix-Variate Beta Probability Distribution on Symmetric Matrices

The notion of generalized power function in the space of real symmetric matrices is used to introduce a kind of extended matrix-variate beta function. With the aid of this, we define a different versions of extended matrix-variate beta distributions. Some fundamental properties of these distributions are established. We show that using a linear transformation on the extended matrix-variate beta distributions of the first and second kind, we can generalize these distributions. We also show that the distribution of the sum of two independent inverse Riesz matrices introduced by Tounsi and Zine (J Multivar Anal 111:174–182, 2012) can be written in terms of the generalized extended matrix-variate beta function. Finally, using Fixed point iterative method, we provide a calculable maximum a posteriori (MAP) estimator for the unknown covariance matrix of a multivariate normal distribution based on the class of the extended matrix-variate beta prior distribution. Additionally, we evaluated the Gaussian finite sample performance by calculating such evaluation criteria as Mean Square Error (MSE) and Hilbert-Schmidt distance (DHS). The obtained results confirm the performance of the proposed prior.

Computing the Maslov index for large systems

We address the problem of computing the Maslov index for large linear symplectic systems on the real line. The Maslov index measures the signed intersections (with a given reference plane) of a path of Lagrangian planes. The natural chart parameterization for the Grassmannian of Lagrangian planes is the space of real symmetric matrices. Linear system evolution induces a Riccati evolution in the chart. For large order systems this is a practical approach as the computational complexity is quadratic in the order. The Riccati solutions, however, also exhibit singularites (which are traversed by changing charts). Our new results involve characterizing these Riccati singularities and two trace formulae for the Maslov index as follows. First, we show that the number of singular eigenvalues of the symmetric chart representation equals the dimension of intersection with the reference plane. Second, the Cayley map is a diffeomorphism from the space of real symmetric matrices to the manifold of unitary symmetric matrices. We show the logarithm of the Cayley map equals the arctan map (modulo 2 i 2\mathrm {i} ) and its trace measures the angle of the Langrangian plane to the reference plane. Third, the Riccati flow under the Cayley map induces a flow in the manifold of unitary symmetric matrices. Using the natural unitary action on this manifold, we pullback the flow to the unitary Lie algebra and monitor its trace. This avoids singularities, and is a natural robust procedure. We demonstrate the effectiveness of these approaches by applying them to a large eigenvalue problem. We also discuss the extension of the Maslov index to the infinite dimensional case.

WHAT IS...a Spectrahedron?

A spectrahedron is a convex set that appears in a range of applications. Introduced in [3], the name “spectra” is used because its definition involves the eigenvalues of a matrix and “-hedron” because these sets generalize polyhedra. First we need to recall some linear algebra. All the eigenvalues of a real symmetric matrix are real and if these eigenvalues are all non-negative then the matrix is positive semidefinite. The set of positive semidefinite matrices is a convex cone in the vector space of real symmetric matrices. A spectrahedron is the intersection of an affinelinear space with this convex cone of matrices. An n-dimensional affine-linear space of real symmetric matrices can be parametrized by

Probabilistic proof of product formulas for Bessel functions

We write, for geometric index values, a probabilistic proof of the product formula for spherical Bessel functions. Though our proof looks elementary in the real variable setting, it has the merit to carry over without any further effort to Bessel-type hypergeometric functions of one matrix argument, thereby avoid complicated arguments from differential geometry. Moreover, the representative probability distribution involved in the last setting is shown to be closely related to the symmetrization of upper-left corners of Haar-distributed orthogonal matrices. Analysis of this probability distribution is then performed and in case it is absolutely continuous with respect to Lebesgue measure on the space of real symmetric matrices, we derive an invariance-property of its density. As a by-product, Weyl integration formula leads to the product formula for Bessel-type hypergeometric functions of two matrix arguments.

On the P-property of Z and Lyapunov-like transformations on Euclidean Jordan algebras

The P, Z, and S properties of a linear transformation on a Euclidean Jordan algebra are generalizations of the corresponding properties of a square matrix on Rn. Motivated by the equivalence of P and S properties for a Z-matrix [2] and a similar result for Lyapunov and Stein transformations on the space of real symmetric matrices [6,5], in this paper, we present two results supporting the conjecture that P and S properties are equivalent for a Z-transformation on a Euclidean Jordan algebra. We show that the conjecture holds for Lyapunov-like transformations and Z-transformations satisfying an additional condition.

Semidefinite programming and matrix scaling over the semidefinite cone

Let E be the Hilbert space of real symmetric matrices with block diagonal form diag( A, M), where A is n× n, and M is an l× l diagonal matrix, with the inner product 〈 x, y〉≡Trace( xy). We assume n+ l⩾1, i.e. allow n=0 or l=0. Given x∈ E, we write x⪰0 ( x≻0) if it is positive semidefinite (positive definite). Let Q: E→ E be a symmetric positive semidefinite linear operator, and μ= min{φ(x)=0.5 Trace(xQx):∥x∥=1,x⪰0} . The problem of testing if μ=0 is a significant problem called Homogeneous Programming. On the one hand the feasibility problem in semidefinite programming (SDP) can be formulated as a Homogeneous Programming problem. On the other hand it is related to the generalization of the classic problem of Matrix Scaling. Let ϵ∈(0,1) be a given accuracy, u= Qe− e, e the identity matrix in E, and N= n+ l. We describe a path-following algorithm that in case μ=0, in O( N ln[N∥u∥/ϵ]) Newton iterations produces d⪰0, ∥ d∥=1 such that φ( d)⩽ ϵ. If μ>0, in O( N ln[N∥u∥/μ]+ ln ln (1/ϵ)) Newton iterations the algorithm produces d≻0 such that ∥ DQDe− e∥⩽ ϵ, where D is the operator that maps w∈ E to d 1/2 wd 1/2. Moreover, we use the algorithm to prove: μ>0, if and only if there exists d≻0 such that DQDe= e, if and only if there exists d≻0 such that Qd≻0. Thus via this duality the Matrix Scaling problem is a natural dual to the feasibility problem in SDP. This duality also implies that in Blum et al. [Bull. AMS 21 (1989) 1] real number model of computation the decision problem of testing the solvability of Matrix Scaling is both in NP and Co-NP. Although the above complexities can be deduced from our path-following algorithm for general self-concordant Homogeneous Programming and for Matrix Scaling obtained in [Scaling dualities and self-concordant homogeneous programming in finite dimensional spaces, Technical Report LCSR-TR-359, Department of Computer Science, Rutgers University, New Brunswick, NJ, 1999], for the problems considered here the present analysis is quite elementary, short, and complete. This simplicity is mainly due to a new inequality derived in this paper that relates the norm of scaled quantities at two successive Newton iterations and implies a quadratic rate of convergence. The present algorithm is not only a simple path-following algorithm for testing the solvability of feasibility problem in SDP, but is also capable of testing solvability of the Matrix Scaling problem. When n=0, the algorithm reduces to the diagonal Matrix Scaling/Linear Programming algorithm of Khachiyan and Kalantari [SIAM J. Optim. 4 (1992) 668]. As in the case of LP the algorithm of this paper can be used to solve the general SDP problem.

Iso-Huygens deformations of the Cayley operator by the general Lagnese-Stellmacher potential

We construct and study iso-Huygens deformations of higher-order hyperbolic differential operators on the space of real symmetric matrices. Using the so-called skew symmetries, we introduce intertwining operators, which enable us to express fundamental solutions for these deformations in an explicit form and obtain conditions under which the strong Huygens principle holds for these deformations.

Invariant Hyperfunction Solutions to Invariant Differential Equations on the Space of Real Symmetric Matrices

The real special linear group of degree n naturally acts on the vector space of n× n real symmetric matrices. How to determine invariant hyperfunction solutions of invariant linear differential equations with polynomial coefficients on the vector space of n× n real symmetric matrices is discussed in this paper. We prove that every invariant hyperfunction solution is expressed as a linear combination of Laurent expansion coefficients of the complex power of the determinant function with respect to the parameter of the power. Then the problem is reduced to the determination of Laurent expansion coefficients.

The propagator of the Calogero-Moser system in an external quadratic potential

We obtain the Hamilton operator of the Calogero-Moser quantum system in an external quadratic potential by conjugating the radial part for the action of SO( n) by conjugacy of the Hamilton operator of the quantum harmonic oscillator on the Euclidean vector space of real symmetric matrices. Then, with Mehler's formula, we derive the propagator of the problem. We also investigate some schemes to change the interaction constant. For two-particle systems, we obtain explicit formulae, whereas for many-particle systems, we reduce the computation of the propagator to finding a definite integral. We give also the short time approximation, the energy levels and the trace of the propagation operator.

Concave gauge functions and applications

Many problems of optimization involve the minimization of an objective function on a convex cone. In this respect we define a concave gauge function which will be used in interior point methods. Application are given in particular on the space of real symmetric matrices.

A stable method for the calculation of the force fields of polyatomic molecules in dependent coordinates

The problem of constructing a stable algorithm for the solution of the inverse vibrational problem in the case of polyatomic molecules (with the use of data on Isotopically substituted compounds) in a complete system of dependent natural coordinates was presented to us. Our formulation of the problem has been presented in [2]. Let Z be the linear space of real symmetric matrices of dimensionality n x n, in which the closeness of the matrices is characterized by the Euclidean norm for vectors of dimensionality n(n + 1)/2 which are composed of the elements lying on or above the principal diagonals of the matrices. Let T-I~Z be the prescribed positive definite symmetric matrix of the kinematic coefficients and let A be an operator which correlates to each matrix U~Z, an ordered (in order of increasing size, for example) set (a vector from the Euclidean space R n) of eigenvalues of the matrix T-*U, i.e., the vector of the squares of the frequencies of the normal vibrations A~R =. The operator A is defined and continuous everywhere on Z. The problem of determin