Geodesic
Spectrally arbitrary complex sign pattern matrices
The electronic journal of linear algebra, Tome 18 (2009), pp. 674-692.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: An n $\times n$ complex sign pattern matrix S is said to be spectrallyarbitraryif for everymonic nth degree polynomial $f(\lambda )$ with coefficients from C, there is a complex matrix in the complex sign pattern class of S such that its characteristic polynomial is $f(\lambda )$. If S is a spectrally arbitrarycomplex sign pattern matrix, and no proper subpattern of S is spectrallyarbitrary, then S is a minimal spectrallyarbitrarycomplex sign pattern matrix. This paper extends the Nilpotent- Jacobian method for sign pattern matrices to complex sign pattern matrices, establishing a means to show that an irreducible complex sign pattern matrix and all its superpatterns are spectrally arbitrary. This method is then applied to prove that for every n $\geq 2$ there exists an n$\times n$ irreducible, spectrallyarbitrarycomplex sign pattern with exactly3 n nonzero entries. In addition, it is shown that every n $\times n$ irreducible, spectrallyarbitrarycomplex sign pattern matrix has at least 3n - 1 nonzero entries.
Classification : 15A18, 05C15
Keywords: complex sign pattern, spectrallyarbitrarypattern, nilpotent 
@article{ELA_2009__18__a9,
     author = {Gao, Yubin and Shao, Yanling and Fan, Yizheng},
     title = {Spectrally arbitrary complex sign pattern matrices},
     journal = {The electronic journal of linear algebra},
     pages = {674--692},
     publisher = {mathdoc},
     volume = {18},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ELA_2009__18__a9/}
}
TY  - JOUR
AU  - Gao, Yubin
AU  - Shao, Yanling
AU  - Fan, Yizheng
TI  - Spectrally arbitrary complex sign pattern matrices
JO  - The electronic journal of linear algebra
PY  - 2009
SP  - 674
EP  - 692
VL  - 18
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ELA_2009__18__a9/
LA  - en
ID  - ELA_2009__18__a9
ER  - 
%0 Journal Article
%A Gao, Yubin
%A Shao, Yanling
%A Fan, Yizheng
%T Spectrally arbitrary complex sign pattern matrices
%J The electronic journal of linear algebra
%D 2009
%P 674-692
%V 18
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ELA_2009__18__a9/
%G en
%F ELA_2009__18__a9
Gao, Yubin; Shao, Yanling; Fan, Yizheng. Spectrally arbitrary complex sign pattern matrices. The electronic journal of linear algebra, Tome 18 (2009), pp. 674-692. http://geodesic.mathdoc.fr/item/ELA_2009__18__a9/