Polynomial stability and potentially stable patterns

Abstract A polynomial (resp. matrix) is stable if all of its roots (resp. eigenvalues) have negative real parts. A sign (resp. nonzero) pattern A is a matrix with entries in { + , − , 0 } (resp. { ⁎ , 0 } ). If there exists a real stable matrix with pattern A , then A is potentially stable. This paper first shows that if p ( t ) = c 0 t n + c 1 t n − 1 + ⋯ + c n is a (real) stable polynomial with c 0 > 0 , then c i c j > c k c l for every 0 ≤ k i ≤ j ≤ n such that i j + k l is even and l = i + j − k ≤ n . Using this, certain patterns are shown to not be potentially stable based solely on the cyclic structure of their digraphs. Next, bounds are given on m n , the minimum number of nonzero entries in an irreducible potentially stable pattern of order n. It is shown that m 8 = 12 and conjectured that m n ≤ ⌈ 3 n / 2 ⌉ for n ≥ 2 . To support this conjecture, a family of irreducible patterns with exactly ⌈ 3 n / 2 ⌉ nonzero entries is described and demonstrated to be potentially stable for small values of n. Finally, the potentially stable nonzero patterns of order at most 4 are characterized.

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

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

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

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

[5]  A note on the Liénard-Chipart criterion and roots of some families of polynomials , 2014, 1407.4852.

[6]  Yi-Young Nie,et al.  New Criteria for Polynomial Stability , 1987 .

[7]  D. Cvetkovic,et al.  Spectra of Graphs: Theory and Applications , 1997 .

[8]  Xuemin Wang,et al.  A Generalization of Xie-Nie Stability Criterion , 2004, IWMM/GIAE.

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

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

[11]  D. Olesky,et al.  Constructions for potentially stable sign patterns , 2012 .

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

[13]  Willem H. Haemers,et al.  Skew-adjacency matrices of graphs , 2012 .

[14]  Juan Manuel Peña,et al.  Almost strict total positivity and a class of Hurwitz polynomials , 2005, J. Approx. Theory.

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

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

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

[18]  Sets of refined inertias of zero–nonzero patterns , 2017 .

[19]  A sufficient condition for a polynomial to be stable , 2008 .

[20]  L. Mirsky,et al.  The Theory of Matrices , 1961, The Mathematical Gazette.