A PROOF OF THE CAYLEY HAMILTON THEOREM
CHRIS BERNHARDT
Let M (n, n) be the set of all n × n matrices over a commutative ring
with identity. Then the Cayley Hamilton Theorem states:
Theorem. Let A ∈ M (n, n) with characteristic polynomial
det(tI − A) = c0 tn + c1 tn−1 + c2 tn−2 + · · · + cn .
Then
c0 An + c1 An−1 + c2 An−2 + · · · + cn I = 0.
In this note we give a variation on a standard proof (see [1], for
example). The idea is to use formal power series to slightly simplify
the argument.
Proof. First, observe that
1
n
det (I − tA) = t det
I − A = c0 + c1 t + c2 t2 + · · · + cn tn .
t
Now Laplace’s formula for calculating the determinant gives the standard equation
det(I − tA)I = (I − tA)adj(I − tA)
where adj(A) denotes the adjugate (or classical adjoint) of A.
If we consider formal power series in t with coefficients in M (n, n)
i i
then I − tA is invertible with Σ∞
0 A t as inverse. So
i i
2
n
(Σ∞
0 A t )(c0 + c1 t + c2 t + · · · + cn t )I = adj(I − tA).
Writing adj(I − tA) as a formal power series in t with coefficients in
M (n, n) gives
i i
∞
i
(c0 + c1 t + c2 t2 + · · · + cn tn )(Σ∞
0 A t ) = Σ0 Bi t .
Observe that the entries in adj(I − tA) are polynomials in t of degree
at most n − 1. So Bi is the zero matrix for i ≥ n. Equating the
coefficients of tn on both sides gives
c0 An + c1 An−1 + c2 An−2 + · · · + cn I = 0.
1
2
CHRIS BERNHARDT
References
[1] G. Birkoff and S. MacLane, A Survey of Modern Algebra, A K Peters, Wellesley,
MA, 1996.
Fairfield University, Fairfield, CT 06824
E-mail address: [email protected]