2004 Student Research Conference:

17th Annual Student Research Conference

17th Annual Student Research Conference

**Minimum Rank of Positive Semi-Definite Matrices with a Prescribed Graph**

Kelly J. Steinmetz

Dr. Sivaram K. Narayan (Central Michigan University) and Dr. Upendra Kulkarni, Faculty Mentors

A complex *n*x*n* matrix *A* = [aij] is said to be *combinatorially symmetric* if for* i* ≠ *j*, aij ≠ 0 implies aji ≠ 0. We associate a simple graph *G* to a combinatorially symmetric matrix *A* such that V(*G*) = {1, 2, …, *n*} and join vertices *i* and *j* if and only if aij ≠ 0. The graph is independent of the diagonal entries of *A*. Define P(*G*) to be the class of all positive semi-definite matrices associated with a given graph *G*. Denote #(*G*) = min {rank *A* | *A* ε P(*G*) } the minimum PSD rank of *G*. In this talk I will present results about the minimum PSD rank of certain classes of graphs, including some very exciting recent results.

**Keywords:** Linear Algebra, Matrices, Eigenvalues, Graph, Symmetric, Semi-Definite, Rank

**Topic(s):**Mathematics

**Presentation Type:** Oral Paper

**Session:** 9-2

**Location:** VH 1232

**Time:** 8:45