Real Positive Semidefinite Matrices Research Articles

A Monte Carlo approach to estimating the spectrum of large positive semidefinite matrices.

The method presented in this paper aims at furnishing a series of numerical values to approximate the matrix eigenvalues. The method is based on a statistical model that involves the first three central moments of the eigenvalue distribution, and it relies on the solution of a nonlinear system of equations that implements the moment-matching method and on a subsequent procedure of Monte Carlo simulations. The method is only applicable to real positive semidefinite matrices (PSD), and it is especially useful when other techniques lead to computational problems, e.g., when the matrices become too large to be processed or the required storage space sets heavy limits to the computational process.

A new lower bound for the positive semidefinite minimum rank of a graph

The real positive semidefinite minimum rank of a graph is the minimum rank among all real positive semidefinite matrices that are naturally associated via their zero-nonzero pattern to the given graph. In this paper, we use orthogonal vertex removal and sign patterns to improve the lower bound for the real positive semidefinite minimum rank determined by the OS-number and the positive semidefinite zero forcing number.

Note on positive semidefinite maximum nullity and positive semidefinite zero forcing number of partial 2-trees

The maximum positive semidefinite nullity of a multigraph G is the largest possible nullity over all real positive semidefinite matrices whose (i,j)th entry (for i 6 j) is zero if i and j are not adjacent in G, is nonzero if fi,jg is a single edge, and is any real number if fi,jg is a multiple edge. The definition of the positive semidefinite zero forcing number for simple graphs is extended to multigraphs; as for simple graphs, this parameter bounds the maximum positive semidefinite nullity from above. The tree cover number T(G) is the minimum number of vertex disjoint induced simple trees that cover all of the vertices of G. The result that M+(G) = T(G) for an outerplanar multigraph G (F. Barioli et al. Minimum semidefinite rank of outerplanar graphs and the tree cover number. Electron. J. Linear Algebra, 22:10-21, 2011.) is extended to show that Z+(G) = M+(G) = T(G) for a multigraph G of tree-width at most 2.

The Cone of Class Function Inequalities for the 4-By-4 Positive Semidefinite Matrices

A function f from the symmetric group Sn into R is called a class function if f(σ) = f(τ) whenever τ is conjugate to σ. Let df be thegeneralized matrix function associated with f, mapping the n-by-n positive semidefinite Hermitian matricesto R. For example, if f(σ) = sgn(σ), then df(A) = det A. We consider the cone Kn of those f for which df(A) ⩾ 0 for all n-by-n positive semidefinite Hermitian matrices. For n = 4 we show that Kn is polyhedral, and explicitly find the extreme rays. Equivalently, f belongs to Kn if df(A) ⩾ 0 for a finite minimal set of ‘test matrices’. This solves a problem posed by Gordon James. As a corollary, we characterize all inequalities involving linear combinations of immanants on the positive semidefinite Hermitian matrices of order n = 4. We obtain similar results for 4-by-4 real positive semidefinite matrices. 1991 Mathematics Subject Classification: 15A15, 15A45, 15A48.

Recent directions in matrix stability

Matrix stability has been intensively investigated in the past two centuries. We review work that has been done in this topic, focusing on the great progress that has been achieved in the last decade or two. We start with classical stability criteria of Lyapunov, Routh and Hurwitz, and Liénard and Chipart. We then study recently proven sufficient conditions for stability, with particular emphasis on P-matrices. We investigate conditions for the existence of a stable scaling for a given matrix. We review results on other types of matrix stability, such as D-stability, additive D-stability, and Lyapunov diagonal stability. We discuss the weak principal submatrix rank property, shared by Lyapunov diagonally semistable matrices. We also discuss the uniqueness of Lyapunov scaling factors, maximal Lyapunov scaling factors, cones of real positive semidefinite matrices and their applications to matrix stability, and inertia preserving matrices.

Cones of real positive semidefinite matrices associated with matrix stability

Three cones of real positive semidefinite matrices are discussed. Characterizations for Lyapunov diagonal semistability and for Lyapunov diagonal near stability of matrices in terms of these cones are obtained. Also, a relation between Lyapunov diagonal semistability and D-stability is established.