Chapter 13 - The mean-downside risk portfolio frontier: A non-parametric approach

Sparse matrix decompositions and graph characterizations

Extreme rays for cones of Hermitian matrix functions that are non-negative on the positive semi-definite matrices

Numerous results are presented that characterize when a complex Hermitian matrix is Birkhoff-James orthogonal, in the trace norm, to a (Hermitian) positive semidefinite matrix or set of positive semidefinite matrices. For example, a simple-to-test criterion that determines which Hermitian matrices are Birkhoff-James orthogonal, in the trace norm, to the set of all positive semidefinite diagonal matrices is developed. Applications in the theory of quantum resources are explored. For example, the quantum states that have modified trace distance of coherence equal to $1$ (the maximal possible value) are characterized, and a connection between the modified trace distance of $2$-entanglement and the NPPT bound entanglement problem is established.

Given an undirected graph G=(V,E) with node set V=[1,n], a subset $S\subseteq V$, and a rational vector $a\in {\rm\bf Q}^{S\cup E}$, the positive semidefinite matrix completion problem consists of determining whether there exists a real symmetric n × n positive semidefinite matrix X=(xij) satisfying xii=ai ($i\in S$) and xij=aij ($ij\in E$). Similarly, the Euclidean distance matrix completion problem asks for the existence of a Euclidean distance matrix completing a partially defined given matrix. It is not known whether these problems belong to NP. We show here that they can be solved in polynomial time when restricted to the graphs having a fixed minimum fill-in, the minimum fill-in of graph G being the minimum number of edges needed to be added to G in order to obtain a chordal graph. A simple combinatorial algorithm permits us to construct a completion in polynomial time in the chordal case. We also show that the completion problem is polynomially solvable for a class of graphs including wheels of fixed length (assuming all diagonal entries are specified). The running time of our algorithms is polynomially bounded in terms of n and the bitlength of the input a. We also observe that the matrix completion problem can be solved in polynomial time in the real number model for the class of graphs containing no homeomorph K4.

In this paper the concept of positive definite bilinear matrix moment functional, acting on the space of all the matrix valued continuous functions defined on a bounded interval [a,b] is introduced. The best approximation matrix problem with respect to such a functional is solved in terms of matrix Fourier series. Basic properties of matrix Fourier series such as the Riemann—Lebesgue, matrix property and the bessel—parseval matrix inequality are proved. The concept of total set with respect to a positive definite matrix functional is introduced, and the totallity of an orthonormal sequence of matrix polynomials with respect to the functional is established.

We consider Hamiltonian matrices obtained by means of symmetric and positive definite matrices and analyse some perturbations that maintain the eigenvalues on the imaginary axis of the complex plane. To obtain this result we prove for such matrices the existence of a diagonal form or, alternatively by means of symplectic transformations, the existence of thesimplestcanonical form. Applications related to a pair of problems in the context of linear algebra and differential equations are also reported. Copyright © 2004 John Wiley & Sons, Ltd.

This work focuses on the class of L -positive semidefinite matrices A such that A A ( L ) ( † ) − A ( L ) ( † ) A is nonsingular, where A ( L ) ( † ) is the generalized Bott-Duffin inverse of A with respect to a subspace L . We give some equivalent characterizations for the nonsingularity of A A ( L ) ( † ) − A ( L ) ( † ) A , A A ( L ) ( † ) + A ( L ) ( † ) A and I n − A ( A ( L ) ( † ) ) 2 A mainly by rank equalities and subspace operations, among which relations are pointed out. Meanwhile we show the inverses of their general versions in a decomposition form. In addition, the continuity of this class of matrices is discussed based on the continuity of the generalized Bott–Duffin inverse.

Norms and inequalities for condition numbers, III

We state theoretical properties for k-means clustering of Symmetric Positive Definite (SPD) matrices, in a non-Euclidean space, that provides a natural and favourable representation of these data. We then provide a novel application for this method, to time-series clustering of pixels in a sequence of Synthetic Aperture Radar images, via their finite-lag autocovariance matrices.

In this paper, in order to improve the convergence rates of iterative method solving the linear system, the improving modified Gauss-Seidel (IMGS) iterative method with a preconditioner is proposed. Some convergence and comparison results are given when is a symmetric definite matrix are provided.

A Note on Solution of the Nearest Correlation Matrix Problem by von Neumann Matrix Divergence

We study the reduction from a bipartite Hermitian matrix H to two single-qubit reduced Hermitian matrices HA and HB and the inverse problem. The inverse problem extends the so-called marginal problem from quantum information theory. We explicitly obtain the expressions of the matrices mentioned above and then extend them to positive semidefinite matrices by showing several inequalities involving a few parameters from the matrices. We also extend our results to multi-quit matrices. Our study constructs the intimate connection between the generalized marginal problems in terms of Hermitian matrices and the traditional marginal problem in terms of positive semidefinite matrices.

The Jordan cononical form of a product of a Hermitian and a positive semidefinite matrix

We prove inequalities on non-integer powers of products of generalized matrices functions on the sum of positive semi-definite matrices. For example, for any real number , positive semi-definite matrices , , and generalized matrix functions such as the determinant and permanent, etc., we have A general scheme is introduced to prove more general inequalities involving m positive semi-definite matrices for that extend the results of other authors.

On the Bures–Wasserstein distance between positive definite matrices