Binomial Coefficient Identity
Posted: Wed Nov 09, 2005 12:28 am
i am assuming that f is defined by a power series and that the operator norm of A is less than the radius of convergence of the power series.
the definition of f(A) is then clear.
ditto for any for any other ops we are considering
define R to be the subset of polynomials h such that h(f(A),g(A)) has the some coefficients in its power series for matrices , as it does for real variables, h(f(x),g(x))
clearly, R contains 1 x and y
R is clearly closed under addition and multiplication (just check the coefficients are defined by the same operations)
so R is all polynomials.
the definition of f(A) is then clear.
ditto for any for any other ops we are considering
define R to be the subset of polynomials h such that h(f(A),g(A)) has the some coefficients in its power series for matrices , as it does for real variables, h(f(x),g(x))
clearly, R contains 1 x and y
R is clearly closed under addition and multiplication (just check the coefficients are defined by the same operations)
so R is all polynomials.