THE CAYLEY-HAMILTON THEOREM

JIM MORROW

The Cayley-Hamilton theorem can be stated as follows.

Theorem 0.1. Let p_{A}(x) be the characteristic polynomial of the n × n matrix A. Then

p_{A}(A) = 0 (0.1)

Cayley stated the theorem in [1] but did not actually give a proof. He said that he verified it for 3 × 3 matrices and that further proof was not necessary. Hamilton apparently verified in [2] that the theorem is true for 3 × 3 matrices. I believe that Frobenius in [3] gave the first full proof of the theorem.

We will prove the theorem in stages. First we notice that if that if A is a diagonal matrix with diagonal entries λ_{1}, …, λ_{n}, then Spec(A) = {λ_{1}, …, λ_{n}}, p_{A}(x) = ∏_{j=1}^{n}(x−λ_{j}), and p_{A}(A) = 0. Next, if A is similar to a diagonal matrix, D = S^{−1}AS, then p_{A} = p_{D} and p_{A}(A) = Sp_{A}(D)S^{−1} = Sp_{D}(D)S^{−1} = 0. Now we have the following lemma.

Lemma 0.1. If the eigenvalues of A are all distinct, then a set of eigenvectors corresponding to different eigenvalues is linear independent and hence A is similar to a diagonal matrix.

Proof. Let {u_{1}, …, u_{n}} is such a set. Suppose that

∑_{1}^{n} c_{j}u_{j} = 0. (0.2)

Of all relations of this type assume that this relation has the smallest number of non-zero coefficients, c_{j}. Suppose that c_{1} is one of these coefficients. From (0.2) it follows that

∑_{1}^{n} c_{j}λ_{j}u_{j} = 0, (0.3)

and hence

∑_{2}^{n} c_{j}(λ_{j} − λ_{1})u_{j} = 0. (0.4)

But this is a relation with exactly one fewer non-zero coefficients. This is a contradiction and the lemma is proved.

It follows from Lemma 0.1 and the paragraph preceding it that every matrix with distinct eigenvalues satisfies its characteristic polynomial.

Now suppose that A is a general matrix. By Schur’s theorem (page 136 of Johnson and Riess) A is similar to an upper triangular matrix T = S^{−1}AS, with diagonal entries d_{1}, …, d_{n}. The diagonal entries are the eigenvalues of T and hence A. With an appropriate choice of the numbers ϵ_{j}, the numbers d_{1} + ϵ_{1}, …, d_{n} + ϵ_{n} are distinct. By changing only these numbers we can produce a matrix, T_{ϵ} which is as close as we desire to T, and hence A_{ϵ} = ST_{ϵ}S^{−1} is as close as we desire to A. But A_{ϵ} has distinct eigenvalues and hence satisfies its characteristic polynomial. This proves that the function f(A) = p_{A}(A) : R^{n^{2}} ↦ R^{n^{2}} is 0 on a dense set of matrices. Since f(A) is continuous on R^{n^{2}}, p_{A}(A) = f(A) = 0 for all A and the Cayley-Hamilton theorem is proved.

References

[1] A. Cayley, “A Memoir on the Theory of Matrices”, Phil. Trans. Royal Society of London 148, 1858, 17-37.

[2] W. R. Hamilton, Lectures on Quaternions, 1853, p. 566.

[3] F. G. Frobenius, “Über lineare Substitutionen und bilineare Formen”, Journal für die Reine und Angewandte Mathematik 84, 1878, 1-63.

Department of Mathematics, University of Washington

   E-mail address: morrow@math.washington.edu