The inverse quadratic eigenvalue problem (IQEP) arises in the field of structural dynamics. It aims to find three symmetric matrices, known as the mass, the damping, and the stiffness matrices, respectively, such that they are closest to the given analytical matrices and satisfy the measured data. The difficulty of this problem lies in the fact that in applications the mass matrix should be pos...

Abstract. The positive semidefinite minimum rank of a simple graph G is defined to be the smallest possible rank over all 1 positive semidefinite real symmetric matrices whose ijth entry (for i 6= j) is nonzero whenever {i, j} is an edge in G and is zero 2 otherwise. The computation of this parameter directly is difficult. However, there are a number of known bounding parameters 3 and technique...

Extending the previous work of Monteiro and Pang (1998), this paper studies properties of fundamental maps that can be used to describe the central path of the monotone nonlinear complementarity problems over the cone of symmetric positive semidefinite matrices. Instead of focusing our attention on a specific map as was done in the approach of Monteiro and Pang (W98), this paper considers a gen...

We show how the zero structure of a basis of the null space of a positive semidefinite matrix can be exploited to very accurately compute its Cholesky factorization. We discuss consequences of this result for the solution of (constrained) linear systems and eigenvalue problems. The results are of particular interest if A and the null space basis are sparse.

We propose an algorithm for solving optimization problems defined on a subset of the cone of symmetric positive semidefinite matrices. This algorithm relies on the factorization X = Y Y T , where the number of columns of Y fixes an upper bound on the rank of the positive semidefinite matrix X. It is thus very effective for solving problems that have a low-rank solution. The factorization X = Y ...

Suppose that A ∈ RN×N is symmetric positive semidefinite with rank K ≤ N . Our goal is to decompose A into K rank-one matrices ∑K k=1 gkg T k where the modes {gk} K k=1 are required to be as sparse as possible. In contrast to eigen decomposition, these sparse modes are not required to be orthogonal. Such a problem arises in random field parametrization where A is the covariance function and is ...