Frank Hall
Professor
Office: 780 COE
Office phone: 404.413.6432
E-mail: matfjh [at]langate.gsu.edu
Personal webpage
Education
B.A. Mathematics, St. Mary's University, 1965
M.S. Mathematics, University of Houston, 1967
Ph.D. Mathematics, North Carolina State University, 1973
Research/Teaching
My past research includes the following topics: generalized inverses, in particular, the fundamental matrix of constrained minimization; integer matrices, including integer LU factorizations; consimilarity of matrices and the conconical form. My more recent research mainly involves qualitative matrix analysis, specifically, sign pattern matrices. In this area we study cyclic, rank, nonsingularity, eigenvalue, stability, and other properties of matrices, based solely on the signs of the entries of the matrices. Motivation for such work comes from fields such as Economics, Energy Planning, and Biology. In research on sign pattern matrices there is an inter-play with various types of graphs, and as such, this area becomes part of combinatorial Matrix Theory. For matrices with complex entries, we study the more general ray pattern matrices. Other recent research involves tournament matrices, graph isomorphisms, and interlacing eigenvalue results for the Normalized Laplacian.
I have taught from the 1000 thru the 8000 level at Georgia State University. Some of my main interests are the three Linear Algebra courses. Recently, I co-developed three new courses: Senior Seminar, Topics in Applied Matrix Analysis, and Combinatorial Matrix Theory. I have been fortunate to have published with some of our graduate students.
Selected Publications
M. Arav, F. J. Hall, Z. Li , A Cauchy-Schwarz Inequality for Triples of Vectors, to appear in Mathematical Inequalities and Applications, Volume 11, 2008.
M. Arav, J. Bevis, F. J. Hall, Inherited LU-Factorizations of Matrices, to appear in Linear Algebra and Its Applications.
F. J. Hall, K. Patel, and M. Stewart, Interlacing Results on Matrices Associated with Graphs, to appear in Journal of Combinatorial Mathematics and Combinatorial Computing.
F. J. Hall and Z. Li, Sign Pattern Matrices, Chapter 33 in Handbook of Linear Algebra, Simon and Hall/CRC Press, 2007.
M. Arav, F. Hall, S. Koyuncu, Z. Li, and B. Rao, Rational realizations of the minimum rank of a sign pattern matrix, Lin. Alg. and Appl. 409: 111-125 (2005).
F. Hall, Z. Li, and J. Stuart, Reducible powerful ray pattern matrices, Lin. Alg. and Appl. 399:125-140 (2005).
G. Chen, G. Davis, F. Hall, Z. Li, , K. Patel, and M. Stewart, An interlacing result on normalized Laplacians, SIAM Journal on Discrete Mathematics, 18:353-361 (2004).
F. Hall, Z. Li, and B. Rao, From Boolean to sign pattern matrices, Lin. Alg. and Appl. 393:233-251 (2004).
F. Hall, and Z. Li, Sign patterns, inverses, and generalized inverse – a brief survey, Numerical Math. J. of Chinese Universities, 12(Supplement), 8-11 (2003).
G. Chen, F. Hall, Z. Li, and B. Wei, On ranks of matrices associated with trees, Graphs and Combinatorics , 19:323-334 (2003).
F. Hall, Z. Li, and J. Stuart, Irreducible powerful ray pattern matrices, Lin. Alg . and Appl . 342:47-58 (2002).
G. Chen, F. Hall, Z. Kezdy, Z. Li, and H. Zhou , Isomorphisms involving reversing arcs of digraphs, The Journal of Combinatorial Mathematics and Combinatorial Computing, 36:155-160 (2001).
F. Hall, Z. Li, and D. Wang, Symmetric sign pattern matrices that require unique inertia, Lin. Alg . and Appl . 338:153-169 (2001).
F. Hall, and Z. Li , Inertia sets of symmetric sign pattern matrices, Numerical Math. J. of Chinese Universities, 10:226-240 (2001).