579 DIFFERENTIAL PROPERTIES OF EIGENVALUES J. Burke, M. Overton, September 1991 We define and study a directional derivative for two functions of the spectrum of an analytic matrix valued function. These are the maximum real part and the maximum modulus of the spectrum. Results are first obtained for the roots of polynomials with analytic coefficients by way of Puiseux-Newton series. In this regard, the primary analytic tool is the so called Puiseux-Newton diagram. These results are then translated into the context of matrices. Precise results are obtained when the eigenvalues that achieve the maximum value for the function under consideration are all either nondefective or nonderogatory. In the defective derogatory cases a general lower bound for the directional derivative is given which, in particular, describes those directions in which the directional derivative attains an infinite value.