The Distance of Potentially Stable Sign Patterns to the Unstable Matrices

Qi, Lin; Olesky, D.D.; Driessche, P. vanden

doi:10.1137/s089547980139087x

Cited by 8 publications

(11 citation statements)

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“…The potentially stable 4 × 4 path and star sign patterns are listed in [6] and [8]. Beginning with these sign patterns, we show that up to equivalence there are exactly 5 path sign patterns and 5 star sign patterns that require H 4 .…”

Section: Tree Sign Patterns If D(a) Is a Doubly Directed Path Then mentioning

confidence: 85%

“…Path sign patterns of order 4. Up to equivalence, the following are the only 4 × 4 path sign patterns that are potentially stable, sign nonsingular, not sign stable, and for which its negative is not potentially stable (see [6] and [8]): Since P 1 , P 2 , P 3 , and P 4 have only nonpositive entries on the diagonal, Theorem 2.3 in [1] applies; i.e., if any one of them allows H 4 , then it also requires H 4 . The following result is immediate from this theorem and the table of realizations below.…”

Section: Tree Sign Patterns If D(a) Is a Doubly Directed Path Then mentioning

confidence: 99%

“…Since −P 5 is not potentially stable [6,8], P 5 does not allow refined inertia (4, 0, 0, 0), and thus by Observation 1.1 it also does not allow 2.2. Star sign patterns of order 4.…”

Section: Realization Of Signmentioning

confidence: 99%

“…In order to obtain a characterization of the 4 × 4 tree sign patterns that require H 4 , we started with the list of potentially stable path and star sign patterns in [6,8]. Elimination of those sign patterns that are sign stable or not sign nonsingular resulted in 21 tree sign patterns P 1 , P 2 , P 3 , P 4 , P 5 and S 1 , S 2 , S 3 , S 4 , S 5 above together with P 6 , P 7 , P 8 , P 9 , P 10 , P 11 and S 6 , S 7 , S 8 , S 9 , S 10 in the Appendix.…”

Section: Realization Of Signmentioning

confidence: 99%

“…. , S 5 , up to equivalence there are eleven 4×4 tree sign patterns from [6] and [8] that are sign nonsingular, potentially stable and not sign stable. We now list these sign patterns and show below that they allow refined inertia (4, 0, 0, 0), and thus do not require H 4 .…”

Section: A Necessary Condition For Requiringmentioning

confidence: 99%

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Refined inertias of tree sign-patterns

Garnett¹,

Olesky²,

Driessche³

2013

ELA

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Abstract. The refined inertia (n + , n − , nz, 2np) of a real matrix is the ordered 4-tuple that subdivides the number n 0 of eigenvalues with zero real part in the inertia (n + , n − , n 0 ) into those that are exactly zero (nz) and those that are nonzero (2np). For n ≥ 2, the set of refined inertias Hn = {(0, n, 0, 0), (0, n − 2, 0, 2), (2, n − 2, 0, 0)} is important for the onset of Hopf bifurcation in dynamical systems. Tree sign patterns of order n that require or allow the refined inertias Hn are considered. For n = 4, necessary and sufficient conditions are proved for a tree sign pattern (necessarily a path or a star) to require H 4 . For n ≥ 3, a family of n × n star sign patterns that allows Hn is given, and it is proved that if a star sign pattern requires Hn, then it must have exactly one zero diagonal entry associated with a leaf in its digraph.

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Section: Tree Sign Patterns If D(a) Is a Doubly Directed Path Then mentioning

confidence: 85%

Section: Tree Sign Patterns If D(a) Is a Doubly Directed Path Then mentioning

confidence: 99%

“…Since −P 5 is not potentially stable [6,8], P 5 does not allow refined inertia (4, 0, 0, 0), and thus by Observation 1.1 it also does not allow 2.2. Star sign patterns of order 4.…”

Section: Realization Of Signmentioning

confidence: 99%

Section: Realization Of Signmentioning

confidence: 99%

Section: A Necessary Condition For Requiringmentioning

confidence: 99%

See 3 more Smart Citations