Functions of matrices

We use the Cayley-Hamilton theorem and the sequence of Horner polynomials associated with a polynomial w(z) to obtain explicit formulas for functions of the form f(tA), where f is defined by a convergent power series and A is a square matrix. We use a well-known explicit formula for the resolvent of A and show that f(tA) is the Hadamard product of f(t) and the function (I − tA)−1, which is easily obtained from the resolvent.