A sufficient condition that a matrix have eigenvalues with non-zero real parts

Abstract

If A is a complex matrix let à be the real matrix obtained by replacing the diagonal elements of A by the moduli of their real parts and by replacing the off-diagonal elements by the negative of their moduli. Then we show that if à is an M-matrix the eigenvalues of A have nonzero real parts and, moreover, the moduli of the real parts are bounded below by the minimum of the real parts of the eigenvalues of Ã.

†Thanks are due to Mr. W. A. Coppel for advice during the preparation of this paper

†Thanks are due to Mr. W. A. Coppel for advice during the preparation of this paper

Notes

†Thanks are due to Mr. W. A. Coppel for advice during the preparation of this paper