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Abstract
Motivated by a work of Boros, Brualdi, Crama and Hoffman, we consider the sets of (i) possible Perron roots of nonnegative matrices with prescribed row sums and associated graph, and (ii) possible eigenvalues of complex matrices with prescribed associated graph and row sums of the moduli of their entries. To characterize the set of Perron roots or possible eigenvalues of matrices in these classes we introduce, following an idea of Al'pin, Elsner and van den Driessche, the concept of row uniform matrix, which is a nonnegative matrix where all nonzero entries in every row are equal. Furthermore, we completely characterize the sets of possible Perron roots of the class of nonnegative matrices and the set of possible eigenvalues of the class of complex matrices under study. Extending known results to the reducible case, we derive new sharp bounds on the set of eigenvalues or Perron roots of matrices when the only information available is the graph of the matrix and the row sums of the moduli of its entries. In the last section of the paper a new constructive proof of the Camion–Hoffman theorem is given.
Original language  English 

Pages (fromto)  187209 
Journal  Linear Algebra and its Applications 
Volume  455 
Early online date  26 May 2014 
DOIs  
Publication status  Published  15 Aug 2014 
Keywords
 Geršgorin
 Eigenvalues
 Perron root
 Row sums
 Row uniform matrices
 Graphs
 Diagonal similarity
 Sum scaling
 Camion–Hoffman
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Dive into the research topics of 'On sets of eigenvalues of matrices with prescribed row sums and prescribed graph'. Together they form a unique fingerprint.Projects
 1 Finished

PerronFrobenius Theory and MaxAlgebraic Combinatorics of Nonnegative Matrices
Butkovic, P.
Engineering & Physical Science Research Council
12/03/12 → 11/03/14
Project: Research Councils