Let A,B be two positive definite matrices. The Fan-Lidskii majorisations can be subsumed as the symmetric norm rearrangement inequalities ‖A↓ +B↑‖ ‖A+B‖ ‖A↓ +B↓‖ where the up/down arrows on A,B mean the diagonal matrices with the same eigenvalues in decreasing/increasing order down to the diagonal. We refine these relations with sums of type A +UBU∗ and A +VBV∗ for two unitary matrices U,V asso...

We consider the positive semidefinite (psd) matrices with binary entries. We give a characterisation of such matrices, along with a graphical representation. We then move on to consider the associated integer polytopes. Several important and well-known integer polytopes — the cut, boolean quadric, multicut and clique partitioning polytopes — are shown to arise as projections of binary psd polyt...

This paper presents a new class of tensor factorization called positive semidefinite tensor factorization (PSDTF) that decomposes a set of positive semidefinite (PSD) matrices into the convex combinations of fewer PSD basis matrices. PSDTF can be viewed as a natural extension of nonnegative matrix factorization. One of the main problems of PSDTF is that an appropriate number of bases should be ...

where B and D are square blocks. We prove the following inequalities for the Schatten q-norm ||.||q , which are sharp when the blocks are of size at least 2× 2: ||A||q ≤ (2 q − 2)||C||q + ||B|| q q + ||D|| q q, 1 ≤ q ≤ 2, and ||A||q ≥ (2 q − 2)||C||q + ||B|| q q + ||D|| q q, 2 ≤ q. These bounds can be extended to symmetric partitionings into larger numbers of blocks, at the expense of no longer...

In this article we apply proper splittings of matrices to develop an iterative process approximate solutions matrix equations the form TX=W. Moreover, by using partial order induced positive semidefinite matrices, obtain equivalent conditions convergence process. We also include some speed comparison results method. addition, for all T propose a splitting based on polar decomposition T.

A univariate trace polynomial is a in variable x and formal symbols Tr ( j ) . Such an expression can be naturally evaluated on matrices, where the are as normalized traces. This paper addresses global constrained positivity of polynomials symmetric matrices all finite sizes. tracial analog Artin's solution to Hilbert's 17th problem given: positive semidefinite quotient sums products squares tr...

Today we continue with some additional aspects of quantifier elimination. We will then recall the Positivstellensatz and its relations with semidefinite programming. After introducing copositive matrices, we present Pólya’s theorem on positive forms on the simplex, and the associated relaxations. Finally, we conclude with an important result due to Schmüdgen about representation of positive pol...

The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI), an affine section of the semidefinite cone, is always dual to the numerical range of a matrix, which is therefore an affine projection of the semidefinite con...

Abstract We present a generalization of the notion neighborliness to non-polyhedral convex cones. Although definition is available in case literature, it fairly restrictive as requires all low-dimensional faces be polyhedral. Our approach more flexible and includes, for example, cone positive-semidefinite matrices special (this not neighborly general). term our Terracini convexity due its conce...