Abstract
A sign pattern is a matrix whose entries are from the set {+,−,0}. Associated with each sign pattern A of order n is a qualitative class of A,defined by Q(A). For a symmetric sign pattern A of order n,the inertia of A is a set i(A)={i(B)=(i +(B),i −(B),i 0(B)) | B=B T H∈Q(A)}, where i + (B) (respectively,i − (B),i 0(B) denotes the number of positive (respectively, negative, zero) eigenvalues. That the symmetric sign pattern A requires unique intertia means i(B 1)=i(B 2) for all real symmetric matrices B 1,B 2∈Q(A). The purpose of this paper is to characterize double star and cycle sign patterns that require unique inertia. Further, their unique inertia is also obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Drew, J. H.,Johnson, C. R.,Olesky, D. D.,et al.,Spectrally arbitrary patterns,Linear Algebra Appl., 2000,308:121–137.
Gao Yubin,Shao Yanling,Inertially arbitrary patterns,Linear and Multilinear Algebra,2001,49(2):161–168.
Hall, F.J.,Li Zhongshan,Wang Di,Symmetric sign pattern matrices that require unique inertia,Linear Algebra Appl.,2001,338:153–169.
Eschenbach, C. A.,Johnson, C. R.,Sign patterns that require real, nonreal and pure imaginary eigenvalues,Linear and Multilinear Algebra,1991,29:299–311.
Brualdi, R. A., Shader, B. L., Matrices of Sign-solvable Linear Systems, Cambridge: Cambridge University Press, 1995.
Horn, R. A.,Johnson, C. R.,Matrix Analysis,Cambridge:Cambridge University Press,1985.
Author information
Authors and Affiliations
Additional information
Supported by Shanxi Natural Science Foundation (20011006).
Rights and permissions
About this article
Cite this article
Shao, Y. Two classes of symmetric sign patterns that require unique inertia. Appl. Math. Chin. Univ. 18, 243–250 (2003). https://doi.org/10.1007/s11766-003-0031-4
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11766-003-0031-4