The real symmetric matrices with a P-set of maximum size and their associated graphs
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摘要: 实对称阵的P-集是一个基于矩阵的特征值重数以及Cauchy插值定理所提出的定义。设 为一个 阶实对称阵，记 为 的特征值0的（代数）重数，并记 为将 的第 行与第 列去掉后所得的主子阵，其中 为 的一个非空子集。特别地，当 时，称S为 的一个P-集。记 为实对称阵 的P-集所含元素个数的最大值。Kim与Shader证明了每个 阶实对称阵至多包含 个元素，即 。杜志斌与Fonseca首先将研究重点放在树矩阵（即伴随图为树的矩阵），研究了满足 的 阶树矩阵 ，并完全刻画出 的伴随图（树）。本文将研究范围从树矩阵延伸到所有实对称阵，研究了满足 的 阶实对称阵 ，给出其相关性质，并对 为偶数时 的伴随图进行特征刻画，而对 为奇数时 的伴随图给出了猜想，推广了关于树矩阵的结果。
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关键词:
- P-集
Abstract: The concept of P-set of real symmetric matrices is proposed based on the multiplicity of eigenvalues of matrices and Cauchy interlacing theory. Suppose that is a real symmetric matrix of order . Let be the multiplicity of eigenvalue 0 of , and let be the principal submatrix of obtained from by deleting the rows and columns, where is a nonempty subset of . In particular, when , we say S is a P-set of . Let be the maximum size among the P-sets of . Kim and Shader proved that every real symmetric matrix of order has at more elements, i.e., . Du and Fonseca first focused on the acyclic matrices (i.e., the matrices whose associated graphs are trees), investigated the acyclic matrices of order n satisfied , and completely characterized the associated graphs (trees) of .This paper extended the research from acyclic matrices to all real symmetric matrices, investigated the real symmetric matrices of order satisfied , presented their properties, and characterized the associated graphs when is even, conjectured the associated graphs when is odd, which improved the results on acyclic matrices.-
Keywords:
- P-set
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[1] Du Z., da Fonseca C.M., The acyclic matrices with a P-set of maximum size[J]. Linear Algebra Appl., 2014, 468:27-37.
[2] Fernande R., da Cruz H.F., Sets of Parter vertices which are Parter sets[J]. Linear Algebra Appl., 2014, 448:37-54.
[3] Horn R.A., Johnson C.R., Matrix Analysis, Second Edition[M]. New York:Cambridge University Press, 2013.
[4] Johnson C.R., Leal Duarte A., Saiago C.M., The Parter-Wiener theorem: refinement and generalization[J]. SIAM J. Matrix Anal. Appl., 2003, 25(2):352-361.
[5] Kim I.-J., Shader B.L., Non-singular acyclic matrices[J]. Linear Multilinear Algebra, 2009, 57(4): 399-407.
[6] Nelson C.G., Shader B.L., All pairs suffice for a P-set [J]. Linear Algebra Appl., 2015, 475:114-118.[1] Du Z., da Fonseca C.M., The acyclic matrices with a P-set of maximum size[J]. Linear Algebra Appl., 2014, 468:27-37.
[2] Fernande R., da Cruz H.F., Sets of Parter vertices which are Parter sets[J]. Linear Algebra Appl., 2014, 448:37-54.
[3] Horn R.A., Johnson C.R., Matrix Analysis, Second Edition[M]. New York:Cambridge University Press, 2013.
[4] Johnson C.R., Leal Duarte A., Saiago C.M., The Parter-Wiener theorem: refinement and generalization[J]. SIAM J. Matrix Anal. Appl., 2003, 25(2):352-361.
[5] Kim I.-J., Shader B.L., Non-singular acyclic matrices[J]. Linear Multilinear Algebra, 2009, 57(4): 399-407.
[6] Nelson C.G., Shader B.L., All pairs suffice for a P-set [J]. Linear Algebra Appl., 2015, 475:114-118.
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