Bounds for characteristic roots of matrices
Olga Taussky · Duke Mathematical Journal · 1948
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