the summation produces the zero matrix. This establishes the Hamilton-Cayley
theorem.
Evaluation of Functions
Let f(x)denote a scalar function of x, where all derivatives with respect to xare
defined at x= 0.Functions which satisfy this condition can then be represented as a
power series expansion about the point x= 0. This power series expansion has the
form
f(x) =∑∞k=0ckxk,where ckare the coefficients of the power series. Let Adenote an n×nmatrix with
characteristic equation C(λ) = 0 which has the roots (eigenvalues) λi,(i= 1, 2 ,.. ., n ).
The infinite power series for f(x)can be represented in the alternative form
f(x) = C(x)∑∞k=0c∗kxk+R(x), (10 .44)where c∗kare new coefficients to be determined, C(x)is the characteristic polynomial
associated with the matrix A, and R(x)is a remainder polynomial of degree less than
or equal to (n−1) which can be expressed in the form
R(x) = β 1 xn−^1 +β 2 xn−^2 +···+βn− 1 x+βnwhere β 1 , β 2 ,... , β nare constants. By the Hamiliton–Cayley theorem C(A) = [0], thus,
the matrix function f(A)becomes
f(A) = R(A), (10 .45)which implies the matrix function f(A)can be expressed as some linear combination
of the matrices {I, A, A^2 ,... , A n−^1 }and consequently must have the form
f(A) = R(A) = β 1 An−^1 +β 2 An−^2 +··· +βn− 1 A+βnIwhere β 1 ,... , β nare constants to be determined. This result is not unexpected since
it has been previously shown how one can use the Hamilton–Cayley theorem to
express all powers of A, greater than or equal to the dimension nof A, in terms of
linear combinations of the integer powers of Aless than or equal to n− 1 .Also, from
equation (10.44), one can write the nspecial relations
f(λi) = R(λi), i = 1, 2 ,... , n, (10 .46)