A note on bounds of multiple characteristic roots of a matrix
P. Stein · Journal of research of the National Bureau of Standards · 1952
If A—((Lij) is an nXn matrix and if Ct are the n circles with centres au and radii 2 3 K« I > Olga, Taussky 3 s=l 89*i proves these two theorems. Theorem A. A characteristic root X, which is an inner or boundary point of only one Ciy cannot have two independent characteristic vectors corresponding to it. Theorem B. If A has a characteristic root X of multiplicity n— 1, with n— 1 independent characteristic vectors, the X lies in at least n— 1 circles C*. In this note it is proved that Theorem C. If X is a characteristic root of A with m<n independent characteristic vectors corresponding to it, then X lies in at least m circles Ct. Theorem C is a generalization of both Theorems A and B and closes the gap between them. Theorem C contains the following generalization of a well-known theorem about determinants (for definitions and references, see, O. Taussky, A recurring theorem on determinants, Am. Math. Monthly 56, 672 (1949)). Theorem D} Let A be a matrix that cannot be