Freedman, Lovász and Schrijver characterized graph parameters that can be represented as the (weighted) number of homomorphisms into a fixed graph. Several extensions of this result have been proved. We use the framework of categories to prove a general theorem of this kind. Similarly as previous resuts, the characterization uses certain infinite matrices, called connection matrices, which are ...

In this paper we present special subsets of positive semidefinite matrices where the linear function κ becomes a geometric similarity and its inverse can be easily computed. The images of these special subsets are characterized geometrically. We also study the systems of coordinates for spherical matrices and at the end, we introduce the class of multibalanced distance

An element A of the n× n copositive cone C is called irreducible with respect to the nonnegative cone N if it cannot be written as a nontrivial sum A = C + N of a copositive matrix C and an elementwise nonnegative matrix N . This property was studied by Baumert [2] who gave a characterisation of irreducible matrices. We demonstrate here that Baumert’s characterisation is incorrect and give a co...

An important property of the Kalman filter is that the underlying Riccati flow is a contraction for the natural metric of the cone of symmetric positive definite matrices. The present paper studies the geometry of a low-rank version of the Kalman filter. The underlying Riccati flow evolves on the manifold of fixed rank symmetric positive semidefinite matrices. Contraction properties of the low-...

X. Zhan has conjectured that the spectral radius of the Hadamard product of two square nonnegative matrices is not greater than the spectral radius of their ordinary product. We prove Zhan’s conjecture, and a related inequality for positive semidefinite matrices, using standard facts about principal submatrices, Kronecker products, and the spectral radius.

We consider the problem of writing an arbitrary symmetric matrix as the difference of two positive semidefinite matrices. We start with simple ideas such as eigenvalue decomposition. Then, we develop a simple adaptation of the Cholesky that returns a difference-of-Cholesky representation of indefinite matrices. Heuristics that promote sparsity can be applied directly to this modification.

The well-known result stating that any non-convex quadratic problem over the nonnegative orthant with some additional linear and binary constraints can be rewritten as linear problem over the cone of completely positive matrices (Burer, 2009) is generalized by replacing the nonnegative orthant with an arbitrary closed convex cone. This set-semidefinite representation result implies new semidefi...

The arithmetic-geometric mean inequality for singular values due to Bhatia and Kittaneh says that 2sj(AB ∗) ≤ sj(A∗A + B∗B), j = 1, 2, . . . for any matrices A,B. We first give new proofs of this inequality and its equivalent form. Then we use it to prove the following trace inequality: Let A0 be a positive definite matrix and A1, . . . , Ak be positive semidefinite matrices. Then tr k ∑

This article aimed at a general audience of computational scientists, surveys the Cholesky factorization for symmetric positive definite matrices, covering algorithms for computing it, the numerical stability of the algorithms, and updating and downdating of the factorization. Cholesky factorization with pivoting for semidefinite matrices is also treated.  2009 John Wiley & Sons, Inc. WIREs Co...

We present a hybrid algorithm for optimizing a convex, smooth function over the cone of positive semidefinite matrices. Our algorithm converges to the global optimal solution and can be used to solve general largescale semidefinite programs and hence can be readily applied to a variety of machine learning problems. We show experimental results on three machine learning problems. Our approach ou...