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INERTIALLYARBITRARYTREESIGNPATTERNSOFORDER 4 ELA

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ELA

INERTIALLY ARBITRARY TREE SIGN PATTERNS OF ORDER 4

∗

YUBIN GAO† AND YANLING SHAO†

Abstract. Ann×nsign pattern matrixAis an inertially arbitrary pattern if for every non- negative triple (n1, n2, n3) withn1+n2+n3=n, there is a real matrix in the sign pattern class ofAhaving inertia (n1, n2, n3). Ann×nsign pattern matrixAis a spectrally arbitrary pattern if for any given real monic polynomialr(x) of degreen, there is a real matrix in the sign pattern class ofAwith characteristic polynomialr(x). In this paper, all 4×4 tree sign pattern matrices that are inertially arbitrary are characterized. As a result, in this paper, it is shown that a 4×4 tree sign pattern matrix is inertially arbitrary if and only if it is spectrally arbitrary.

Key words. Sign pattern matrix, Inertially arbitrary pattern, Spectrally arbitrary pattern, Tree sign pattern.

AMS subject classifications. 15A18, 15A29.

∗Received by the editors on July 18, 2011. Accepted for publication on November 20, 2011.

Handling Editor: Michael Tsatsomeros.

†Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, P.R. China ([email protected], [email protected]). Research supported by NNSF of China (no. 11071227).

Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 22, pp. 1148-1155, November 2011

http://math.technion.ac.il/iic/ela

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