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Keywords:
sign pattern; inertia; inertia set; unique inertia
sign pattern; inertia; inertia set; unique inertia
Summary:
The inertia set of a symmetric sign pattern $A$ is the set $i(A)=\lbrace i(B) \mid B=B^T \in Q(A)\rbrace $, where $i(B)$ denotes the inertia of real symmetric matrix $B$, and $Q(A)$ denotes the sign pattern class of $A$. In this paper, a complete characterization on the inertia set of the nonnegative symmetric sign pattern $A$ in which each diagonal entry is zero and all off-diagonal entries are positive is obtained. Further, we also consider the bound for the numbers of nonzero entries in the nonnegative symmetric sign patterns $A$ with zero diagonal that require unique inertia.
References:
[1] B. N. Datta: Stability and inertia. Linear Algebra Appl. 302–303 (1999), 563–600. MR 1733550 | Zbl 0972.15009
[2] J. H. Drew, C. R. Johnson, D. D. Olesky and P. van den Driessche: Spectrally arbitrary patterns. Linear Algebra Appl. 308 (2000), 121–137. MR 1751135
[3] R. A. Horn and C. R. Johnson: Matrix Analysis. Cambridge University Press, Cambridge, 1985. MR 0832183
[4] R. A. Brualdi and B. L. Shader: Matrices of Sign-solvable Linear System. Cambridge University Press, Cambridge, 1995. MR 1358133
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