Bounds for characteristic roots of matrices
Bounds for characteristic roots of matrices
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矩阵特征根的界限
DOI：
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发表时间：
1948
期刊：
影响因子：
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通讯作者：
O. Taussky
中科院分区：
文献类型：
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作者：
O. Taussky
它的根是0 -3 5。三个圆都包含根0。然而，可以证明，如果对元素aik施加进一步的条件，则n = 2的情况类似。定理1。一个正元素矩阵的优势根不可能是所有n个圆i的公共点，除非它是至少两个圆的公共边界点。证明。已知[2]，该矩阵的优势根A是实正的，对应的特征向量XI，…， Xn的选择可以使它的所有分量都是正的。考虑这个方程
which has the roots 0, -3, 5. The root 0 is contained in all three circles. It can however be shown that an analogue of the situation for n = 2 holds if further conditions are imposed on the elements aik. Theorem 1. The dominant root of a matrix of positive elements canno t be a common point of all n circles Oi unless it is a common boundary point of at least two of the circles. Proof. It is known [2] that the dominant root A of such a matrix is real and positive and that the corresponding characteristic vector XI, ... , Xn can be chosen in such a way that all its components are posi tive. Consider then the equation