abs_vol18_pp146_161. 24KB Jun 04 2011 12:05:46 AM

ELA

TOTALLY POSITIVE COMPLETIONS FOR MONOTONICALLY

LABELED BLOCK CLIQUE GRAPHS

∗

CHARLES R. JOHNSON† _{AND CRIS NEGRON}‡

Abstract. It is shown that if the connected graph of the specified entries of a combinatorially symmetric, partial totally positive matrix is monotonically labeled block clique, then there is a totally positive completion. Necessarily the completion strategy is very different from and more complicated than the known totally nonnegative one. The completion preserves symmetry and can be used to solve some non-connected or rectangular, nonsymmetric completion problems.

Key words.Totally positive matrix completion problems, Partial matrix, Monotonically labeled block clique graph.

AMS subject classifications.15A15, 15A57, 05C50.

∗ _{Received by the editors September 24, 2007. Accepted for publication February 24, 2009.}

Handling Editor: Ravindra B. Bapat. This research was supported by NSF grant DMS-03-53510.

†_{Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187, USA}

(crjohnso@math.wm.edu).

‡_{Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903, USA}

(cn9c@virginia.edu).

Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 18, pp. 146-161, March 2009