A bar framework G(p) in r-dimensional Euclidean space is a graph G = (V,E) on the vertices 1, 2, . . . , n, where each vertex i is located at point p in R. Given a framework G(p) in R, a problem of great interest is that of determining whether or not there exists another framework G(q), not obtained from G(p) by a rigid motion, such that ||q−q || = ||p−p || for all (i, j) ∈ E. This problem is k...

In this paperwe study semidefinite programming (SDP)models for a class of discrete and continuous quadratic optimization problems in the complex Hermitian form. These problems capture a class of well-known combinatorial optimization problems, as well as problems in control theory. For instance, they include theMAX-3-CUT problem where the Laplacian matrix is positive semidefinite (in particular,...

The nonnegative and positive semidefinite (PSD-) ranks are closely connected to the nonnegative and positive semidefinite extension complexities of a polytope, which are the minimal dimensions of linear and SDP programs which represent this polytope. Though some exponential lower bounds on the nonnegative [FMP12] and PSD[LRS15] ranks has recently been proved for the slack matrices of some parti...

Unless specified otherwise, all vectors in this lecture live in Rn, and all matrices are symmetric and live in Rn×n. For two vectors v,w, let v · w = ∑ i viwi denote their inner product, and v 0 indicate that all vi ≥ 0. For two matrices A and B, denote by A •B their inner product thinking of them as vectors in Rn2 , i.e. A • B = ∑ ij AijBij = Tr(A >B). Here Tr(·) denotes the trace of a matrix....

Given a n × n positive semidefinite matrix A and a subspace S of C, Σ(S, A) denotes the shorted matrix of A to S. We consider the notion of spectral shorted matrix ρ(S, A) = lim m→∞ Σ(S, A). We completely characterize this matrix in terms of S and the spectrum and the eigenspaces of A. We show the relation of this notion with the spectral order of matrices and the Kolmogorov’s complexity of A t...

Positive semidefinite rank (PSD-rank) is a relatively new quantity with applications to combinatorial optimization and communication complexity. We first study several basic properties of PSD-rank, and then develop new techniques for showing lower bounds on the PSD-rank. All of these bounds are based on viewing a positive semidefinite factorization of a matrix M as a quantum communication proto...

The polar decomposition of an m x n matrix A of full rank, where rn n, can be computed using a quadratically convergent algorithm of Higham SIAMJ. Sci. Statist. Comput., 7 (1986), pp. 1160-1174]. The algorithm is based on a Newton iteration involving a matrix inverse. It is shown how, with the use of a preliminary complete orthogonal decomposition, the algorithm can be extended to arbitrary A. ...

Correlation clustering is a technique for aggregating data based on qualitative information about which pairs of objects are labeled ‘similar’ or ‘dissimilar.’ Because the optimization problem is NP-hard, much of the previous literature focuses on finding approximation algorithms. In this paper we explore how to solve the correlation clustering objective exactly when the data to be clustered ca...

It is shown that a matrix satisfying a certain spectral condition which has an infinite sequence of accretive powers is unitarily similar to the direct sum of a normal matrix and a nilpotent matrix. If the sequence of exponents is forcing or semiforcing then the spectral condition is automatically satisfied . If, further, the index of 0 as an eigenvalue of A is at most 1 or the first term of th...