Stability analysis of diagonally equipotent matrices

Diagonally equipotent matrices are diagonally dominant matrices for which dominance is never strict in any coordinate. They appear e.g. as Laplacian matrices of signed graphs. We show in this paper that for this class of matrices it is possible to provide a complete characterization of the stability properties based only on the signs of the entries of the matrices.

[1]  Daniel Hershkowitz,et al.  Lyapunov Diagonal Semistability of Real H-Matrices , 1985 .

[2]  Richard A. Brualdi,et al.  Matrices of Sign-Solvable Linear Systems , 1995 .

[3]  Gernot M. Engel,et al.  Cyclic and diagonal products on a matrix , 1973 .

[4]  G Iacono,et al.  Determining the distance to monotonicity of a biological network: a graph-theoretical approach. , 2010, IET systems biology.

[5]  Frank J. Hall,et al.  Sign Pattern Matrices , 2006 .

[6]  Magnus Egerstedt,et al.  Graph Theoretic Methods in Multiagent Networks , 2010, Princeton Series in Applied Mathematics.

[7]  O. Taussky.  A Recurring Theorem on Determinants , 1949 .

[8]  Vlastimil Pták,et al.  Cyclic products and an inequality for determinants , 1969 .

[9]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[10]  Dragoslav D. Šiljak,et al.  Large-Scale Dynamic Systems: Stability and Structure , 1978 .

[11]  Tin-Zhu Huang.  Stability Criteria for Matrices , 1998, Autom..

[12]  Claudio Altafini,et al.  Consensus Problems on Networks With Antagonistic Interactions , 2013, IEEE Transactions on Automatic Control.

[13]  L. Kolotilina.  Nonsingularity/singularity criteria for nonstrictly block diagonally dominant matrices , 2003 .

[14]  D. Siljak,et al.  On stability of interval matrices , 1994, IEEE Trans. Autom. Control..

[15]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[16]  Anders Rantzer,et al.  Distributed control of positive systems , 2011, IEEE Conference on Decision and Control and European Control Conference.

[17]  Hal L. Smith.  Systems of ordinary differential equations which generate an order preserving flow. A survey of results , 1988 .

[18]  Kenneth H. Rosen,et al.  Discrete Mathematics and its applications , 2000 .

[19]  Huang Tin-Zhu.  Stability criteria for matrices , 1998 .

[20]  Daniel Hershkowitz,et al.  Recent directions in matrix stability , 1992 .

[21]  Thomas Zaslavsky.  Signed graphs , 1982, Discret. Appl. Math..

[22]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[23]  E. Kaszkurewicz,et al.  Matrix diagonal stability in systems and computation , 1999 .

[24]  J. Willems.  Lyapunov Functions for Diagonally Dominant Systems , 1975 .

[25]  Claudio Altafini,et al.  Achieving consensus on networks with antagonistic interactions , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[26]  J. Cortés,et al.  Distributed strategies for making a digraph weight-balanced , 2009, 2009 47th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[27]  Arjan van der Schaft,et al.  Balanced chemical reaction networks governed by general kinetics , 2012 .

[28]  K. Binder,et al.  Spin glasses: Experimental facts, theoretical concepts, and open questions , 1986 .

[29]  F. Harary,et al.  STRUCTURAL BALANCE: A GENERALIZATION OF HEIDER'S THEORY1 , 1977 .

[30]  R. Bru,et al.  Classes of general H-matrices , 2008 .

[31]  Wei Ren,et al.  Information consensus in multivehicle cooperative control , 2007, IEEE Control Systems.

[32]  James Quirk,et al.  Qualitative Problems in Matrix Theory , 1969 .

[33]  R. Brualdi.  Matrices eigenvalues, and directed graphs , 1982 .