Fausto Bernardini, Holly Rushmeier
Proceedings of SPIE - The International Society for Optical Engineering
Brualdi brought to Geršgorin Theory the concept that the digraph G(A) of a matrix A is important in studying whether A is singular. He proved, for example, that if, for every directed cycle of G(A), the product of the diagonal entries exceeds the product of the row sums of the moduli of the off-diagonal entries, then the matrix is nonsingular. We will show how, in polynomial time, that condition can be tested and (if satisfied) produce a diagonal matrix D, with positive diagonal entries, such that AD (where A is any nonnnegative matrix satisfying the conditions) is strictly diagonally dominant (and so, A is nonsingular). The same D works for all matrices satisfying the conditions. Varga raised the question of whether Brualdi's conditions are sharp. Improving Varga's results, we show, if G is scwaltcy (strongly connected with at least two cycles), and if the Brualdi conditions do not hold, how to construct (again in polynomial time) a complex matrix whose moduli satisfy the given specifications, but is singular. © 2007 Elsevier Inc. All rights reserved.
Fausto Bernardini, Holly Rushmeier
Proceedings of SPIE - The International Society for Optical Engineering
Hang-Yip Liu, Steffen Schulze, et al.
Proceedings of SPIE - The International Society for Optical Engineering
Hannaneh Hajishirzi, Julia Hockenmaier, et al.
UAI 2011
John R. Kender, Rick Kjeldsen
IEEE Transactions on Pattern Analysis and Machine Intelligence