and continuing in this manner, one finds the general form
R(A) = f(A) = Ak=β 1 A+β 2 I
for some constants β 1 and β 2 (i.e., f(A)is some linear combination of {I, A }).For this
example,
R(x) = β 1 x+β 2 and f(x) = xk
and consequently
R(λ 1 ) = R(1) = β 1 +β 2 = (1)k= 1 = f(1)
R(λ 2 ) = R(5) = 5β 1 +β 2 = (5)k=f(5).
(7 .11)
From these equations the unknown constants β 1 and β 2 can be determined. As an
exercise show that
β 1 =
1
4 (5
k−1) and β 2 =^1
4 (5 −^5
k).
The matrix relation
R(A) = f(A) = Ak=
(
5 k− 1
4
)
A+
(
5 − 5 k
4
)
I
is a general formula for expressing the powers of the matrix Aas a linear combination
of the matrices {A, I }. Checking this result with the previous calculations obtained
by use of the Hamilton–Cayley theorem, one finds
for k= 0, A^0 =I
for k= 1, A^1 =A
for k= 2, A^2 = 6A− 5 I
for k= 3, A^3 = 31 A− 30 I
for k= 4, A^4 = 156A− 155 I
which agrees with the previous results.
In general, the Hamilton-Cayley theorem implies that if A is a n×n square
matrix, then powers of A, say Am, for integer values of m, can be represented in the
form
Am=c 0 I+c 1 A+c 2 A^2 +···+cn− 1 An−^1
where c 0 ,... , cn− 1 are constants to be determined.
