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標題: 4x4 符號對稱且三對角的代數正矩陣之結構
4x4 Symmetric Tridiagonal Sign Patterns which Require Algebraic Positivity
作者: 羅佛亞
Loranty Folia Simanjuntak
關鍵字: 符號矩陣;代數正性質;Sign Pattern;Algebraic Positivity
引用: References [1] E.J.Barbeau, Problem books in Mathematics Polynomials, Springer, New York, 1989. [2] A.Berman, M.Catral, L.M.Dealba, A.Elhashash and F.J.Hall, Sign pattern that allow eventual positivity, Electronic journal of algebra, 19 (2010), 108-120. [3] R. A. Brualdi and B. L. Shader, Matrices of Sign-Solvable Linear System, Cambridge University Press, Cambridge, 1995. [4] F.Chatelin, Eigenvalues of Matrices, John Wiley and Sons, Inc, 1993. [5] N.Erawati and A.B.Panrita, Perluasan teorema Cayley-Hamilton pada matriks m xn, Jurnal Matematika, Statistika dan Komputasi, 8 (2011), 1-11. [6] C.A.Eshenbach, F.J.Hall and Z.Li, Some sign patterns that allow a real inverse pair B and B^{-1}, Linear Algebra and its Applications, 252 (1997), 299-321. [7] D.C. Falvo and R.Larson, Elementary Linear Algebra (Sixth Edition), Houghton Mi in Harcourt Publishing Company, Boston New York, 2009. [8] F.R. Gantmacher, Applications of The Theory of Matrices, Chelsea Publishing Company, New York, 1959. [9] F.R. Gantmacher, The Theory of Matrices, Chelsea Publishing Company, New York, 1960. [10] M.R. Hestenes and E. Landesman, Linear Algebra for Mathematics, Science, and Engineering, Prentice Hall, California 2011. [11] P.J. Kim, On the 4 x4 Irreducible Sign Pattern Matrices that Require Four Distinct Eigenvalues, Thesis, Georgia State University, August 2011. [12] S. Kirkland, P. Qiau and X. Zhan, Algebraically positive matrices, Linear Algebra and its Applications, 504 (2016), 14-26. [13] D. Poole, Linear Algebra a modern introduction, Richard Stratton, Canada, 2003. [14] P. A. Samuelson, Foundations of Economic Analysis, Harvard University Press, Cambridge, Mass., 1947; New York, 1971.
本論文中,我們探討了具有代數正性質的 4x4 三對角符號對稱的符號矩陣。
刻劃了此類型 4x4 符號矩陣要求(require) 代數正性質的符號矩陣表現式。

A sign pattern (or sign pattern matrix) is a matrix whose entries come from the set ${-,0,+}$. For a real matrix $A$, $sgn(A)$ is the sign pattern of $A$ whose entries are the signs of the corresponding entries in $A$. If $mathcal{A}$ is a sign pattern, the sign pattern class of $mathcal{A}$ is denoted by Q($mathcal{A}$), which is the set of all real matrices $A$ with $sgn(A)=mathcal{A}$. If $mathcal{P}$ is a property referring to a real matrix, then a sign pattern $mathcal{A}$ $requires$ $mathcal{P}$ if every real matrix in Q($mathcal{A}$) has property $mathcal{P}$. A positive matrix, $A>0$, is a matrix all of whose entries are positive real numbers. A square real matrix $A$ is said to be algebraically positive if there exists a real polynomial $p$ such that $p(A)$ is a positive matrix.

In this thesis, we study the sign patterns that require algebraic positivity. We give a characterization of $4 imes 4$ symmetric tridiagonal sign patterns that require algebraic positivity.
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