In this paper, we study the semidefinite inverse eigenvalue problem of reconstructing a real n-by-n matrix C such that it is nearest to the original pre-estimated real n-by-n matrix Co in the Frobenius norm and satisfies the measured partial eigendata, where the required matrix C should preserve the symmetry, positive semidefiniteness, and the prescribed entries of the preestimated matrix Co. W...

We study a smoothing Newton method for solving a nonsmooth matrix equation that includes semidefinite programming and the semidefinite complementarity problem as special cases. This method, if specialized for solving semidefinite programs, needs to solve only one linear system per iteration and achieves quadratic convergence under strict complementarity and nondegeneracy. We also establish quad...

Let Ln be the n-dimensional second order cone. A linear map from Rm to Rn is called positive if the image of Lm under this map is contained in Ln. For any pair (n,m) of dimensions, the set of positive maps forms a convex cone. We construct a linear matrix inequality (LMI) that describes this cone. Namely, we show that its dual cone, the cone of Lorentz-Lorentz separable elements, is a section o...

Let A = (aij) be an n × n symmetric matrix with all positive entries and just one positive eigenvalue. Bapat proved then that the Hadamard inverse of A, given by A = ( 1 aij ) is positive semidefinite. We show that if moreover A is invertible then A is positive definite. We use this result to obtain a simple proof that with the same hypotheses on A, except that all the diagonal entries of A are...

Let K ⊂ E, K′ ⊂ E′ be convex cones residing in finite-dimensional real vector spaces. An element y in the tensor product E ⊗ E′ is K ⊗ K′-separable if it can be represented as finite sum y = P l xl ⊗ x ′ l, where xl ∈ K and x ′ l ∈ K ′ for all l. Let S(n), H(n), Q(n) be the spaces of n × n real symmetric, complex hermitian and quaternionic hermitian matrices, respectively. Let further S+(n), H+...

In this paper we study several issues related to the characterization of specific classes of multivariate quadratic mappings that are nonnegative over a given domain, with nonnegativity defined by a pre-specified conic order. In particular, we consider the set (cone) of nonnegative quadratic mappings defined with respect to the positive semidefinite matrix cone, and study when it can be represe...

According to a celebrated result by Löwner, a real-valued function f is operator monotone if and only if its Löwner matrix, which is the matrix of divided differences Lf = ( f(xi)−f(xj) xi−xj )N i,j=1 , is positive semidefinite for every integer N > 0 and any choice of x1, x2, . . . , xN . In this paper we answer a question of R. Bhatia, who asked for a characterisation of real-valued functions...