On the equation 𝑃(𝐴,𝑋)=0 in matrices
William E. Roth · Transactions of the American Mathematical Society · 1933
t Considerations which follow do not require the functions ii(A) (k = 0, 1, • • • , p) to be polynomials.In fact any functions which, together with at most their first n-1 derivatives, may be expanded into series of non-negative powers of X are permissible, provided that the characteristic values of A lie within or on the circles of convergence of each of the series representing Ft(X) (k = 0,1, • • • ,p) and their first »-1 derivatives.For information on such functions of matrices the reader may consult Hensel, Über Potenzenreihen von Matrken, Journal für die reine und angewandte Mathematik, vol.155, pp.107-110; Sheffer, A note on matrix power series, American Mathematical Monthly, vol.36 (1930), pp.228-231.