Allow problems concerning spectral properties of sign pattern matrices: A survey

Abstract An n × n sign pattern matrix has entries in { + , - , 0 } . This paper surveys the following problems concerning spectral properties of sign pattern matrices: sign patterns that allow all possible spectra (spectrally arbitrary sign patterns); sign patterns that allow all inertias (inertially arbitrary sign patterns); sign patterns that allow nilpotency (potentially nilpotent sign patterns); and sign patterns that allow stability (potentially stable sign patterns). Relationships between these four classes of sign patterns are given, and several open problems are identified.

[1]  On the Spectra of Striped Sign Patterns , 2003 .

[2]  Judith J. McDonald,et al.  Sign-patterns which require a positive eigenvalue , 1996 .

[3]  Dmitriĭ Olegovich Logofet,et al.  Matrices and Graphs Stability Problems in Mathematical Ecology , 1993 .

[4]  Charles R. Johnson,et al.  Sign patterns that require repeated eigenvalues , 1993 .

[5]  C. Ballantine,et al.  Stabilization by a diagonal matrix , 1970 .

[6]  Inertias of zero–nonzero patterns , 2007 .

[7]  Yubin Gao,et al.  On the potential stability of star sign pattern matrices , 2001 .

[8]  Two classes of symmetric sign patterns that require unique inertia , 2003 .

[9]  P. Samuelson,et al.  Foundations of Economic Analysis. , 1948 .

[10]  Spectrally arbitrary ray patterns , 2008 .

[11]  D. Olesky,et al.  Spectrally arbitrary patterns , 2000 .

[12]  Frank J. Hall,et al.  Eigenvalue frequency and consistent sign pattern matrices , 1994 .

[13]  Potentially nilpotent sign pattern matrices , 1999 .

[14]  Charles R. Johnson.  Combinatorial matrix analysis: An overview , 1988 .

[15]  L. Elsner,et al.  On the spectra of close-to-Schwarz matrices , 2003 .

[16]  Qing Lin,et al.  The Distance of Potentially Stable Sign Patterns to the Unstable Matrices , 2002, SIAM J. Matrix Anal. Appl..

[17]  Yubin Gao,et al.  Sign patterns allowing nilpotence of index 3 , 2007 .

[18]  Charles R. Johnson,et al.  Nested sequences of principal minors and potential stability , 1997 .

[19]  T. Bone.  Positive Feedback May Sometimes Promote Stability , 1983 .

[20]  Victor Klee,et al.  Qualitative stability of linear systems , 1987 .

[21]  L. Hogben.  Handbook of Linear Algebra , 2006 .

[22]  Frank J. Hall,et al.  Symmetric sign pattern matrices that require unique inertia , 2001 .

[23]  James P. Quirk,et al.  Qualitative Economics and the Stability of Equilibrium , 1965 .

[24]  D. Olesky,et al.  Inertially arbitrary sign patterns with no nilpotent realization , 2007 .

[25]  Inertia sets of symmetric 2-generalized star sign patterns , 2006 .

[26]  Satoru Iwata,et al.  Computing the inertia from sign patterns , 2007, Math. Program..

[27]  Yubin Gao,et al.  Inertially arbitrary patterns , 2001 .

[28]  Takeo Yamada,et al.  Generic Matrix Sign-Stability , 1987, Canadian Mathematical Bulletin.

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

[30]  Frank,et al.  INERTIA SETS OF SYMMETRIC SIGN PATTERN MATRICES , 2001 .

[31]  Charles R. Johnson,et al.  A combinatorial converse to the Perron-Frobenius theorem , 1990 .

[32]  Victor Klee,et al.  Linear algorithms for testing the sign stability of a matrix and for findingZ-maximum matchings in acyclic graphs , 1977 .

[33]  Pauline van den Driessche,et al.  Minimal Spectrally Arbitrary Sign Patterns , 2004, SIAM J. Matrix Anal. Appl..

[34]  Sign structures of 3 × 3 stable matrices and their generalization to higher‐order matrices , 1988 .

[35]  W. Greub.  Linear Algebra , 1981 .

[36]  Charles R. Johnson,et al.  Some sign patterns that preclude matrix stability , 1988 .

[37]  Charles R. Johnson,et al.  The potentially stable tree sign patterns for dimensions less than five , 1989 .

[38]  Victor Klee,et al.  Sign-Patterns and Stability , 1989 .

[39]  Sign pattern matrices that allow a nilpotent matrix , 1996, Bulletin of the Australian Mathematical Society.

[40]  A. T. Fuller,et al.  On the stabilization of matrices and the convergence of linear iterative processes , 1958, Mathematical Proceedings of the Cambridge Philosophical Society.

[41]  C. Eschenbach.  Sign patterns that require exactly one real eigenvalue and patterns that require n−1 nonreal eigenvalues , 1993 .

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

[43]  Chi-Kwong Li,et al.  MAPS PRESERVING SPECTRAL RADIUS, NUMERICAL RADIUS, SPECTRAL NORM ∗ , 2007 .

[44]  Charles R. Johnson,et al.  Sign patterns that require real, nonreal or pure imaginary eigenvalues , 1991 .

[45]  Richard A. Brualdi,et al.  Combinatorial matrix theory , 1991 .

[46]  Spectrally arbitrary patterns: Reducibility and the 2n conjecture for n = 5 , 2007 .

[47]  P. D. Driessche,et al.  Spectrally arbitrary star sign patterns , 2005 .

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

[49]  D. D. Olesky,et al.  Low rank perturbations and the spectrum of a tridiagonal sign pattern , 2003 .