ON THE BOUNDS OF THE EIGENVALUES OF MATRIX POLYNOMIALS

Let P(z) := of degree n, such that Xn j=0 Ajzj, Aj E Cmxm, 0 < j < n be a matrix polynomial An & GE;An-1 & GE;... & GE; A0 & GE; 0, An> 0. Then the eigenvalues of P(z) lie in the closed unit disk. This theorem proved by Dirr and Wimmer [IEEE Trans. Automat. Control 52(2007), 2151-2153 ] is infact a matrix extension of a famous and elegant result on the distribution of zeros of polynomials known as Enestrom-Kakeya theorem. In this paper, we prove a more general result which inter alia includes the above result as a special case. We also prove an improvement of a result due to Le<SIC>, Du, Nguye<SIC>n [Oper. Matrices, 13(2019), 937-954] besides a matrix extention of a result proved by Mohammad [Amer. Math. Monthly, vol.74, No.3, March 1967].