Cayley-Hamilton theorem on square matrices

Question

Can anyone help me by giving the proof of the Cayley-Hamilton theorem? It states that every square matrix $A$ satisfies its own characteristic equation:

$p_{A}(A)=0$.

I could prove it when $A$ has distinct eigenvalues, because then it will have $n$ linearly independent eigenvectors, but I couldn't prove the general case.

Answer 1

There are many, many ways to prove the Cayley-Hamilton theorem, and many of them have something to offer in the way of simplicity, generality, or of providing insight into why such a result could be expected to hold in the first place. For instance the case of distinct eigenvalues you could prove can be seen to suffice, once one realises that $p_{A}(A)=0\in\mathrm{Mat}_n(\mathbf C)$ is just a (tremendously complicated) system of $n^2$ polynomial equations in the coefficients of $A$, each of which will hold everywhere once it holds on the dense subset of matrices with $n$ distinct eigenvalues.

Personally I find the following approach leads most naturally to the Cayley-Hamilton theorem, i.e., it emerges quite logically from general considerations that are anyway central to eigenvalue problems, rather than being a claim coming out of the blue.

Certainly (by finite dimensionality of the matrix space) there exist nonzero polynomials $P$ with $P(A)=0$, and by a standard Euclidean division argument there is a polynomial $\mu_A$ such that this happens exactly when $P$ is a (polynomial) multiple of $\mu_A$, the minimal polynomial of $A$. An easy but crucial observation is $(*)$: for any polynomial $D$ dividing $\mu_A$, the quotient $\mu_A/D$ is the minimal polynomial of the restriction of (the linear map defined by) $A$ to the subspace that is the image of $D(A)$. From this it follows that a degree $1$ unitary polynomial $X-\lambda$ divides $\mu_A$ if and only if $\lambda$ is an eigenvalue of $A$. Also if $m$ is the multiplicity if $X-\lambda$ as (irreducible) factor of $\mu_A$, then the image of $(A-\lambda I_n)^m$, having no eigenvalue $\lambda$, forms a direct sum with its kernel, which is by definition the characteristic subspace (or generalised eigenspace) for $\lambda$, and the direct sum is the whole space by the rank-nullity theorem. By induction on the number of eigenvalues, the space then decomposes as direct sum of characteristic subspaces (provided $\mu_A$ splits into linear factors, as certainly happens over an algebraically closed fields like the complex numbers).

All this is standard stuff about the minimal polynomial. Visibly it has the same roots as the characteristic polynomial $p_A$ (namely all eigenvalues), and it is natural to compare the multiplicities. The multiplicity of $\lambda$ as root of $\mu_A$ is the number $m$ above, and its multiplicity as a root of $p_A$ is the dimension of the characteristic subspace for $\lambda$. By $(*)$, the powers $(A-\lambda I_n)^i$ for $i=0,1,\ldots,m$ must all have distinct (ever smaller) images, and therefore also all have distinct (ever larger) kernels, so the dimension of the largest of these kernels, the characteristic subspace for $\lambda$, is at least $m$. This means that (assuming an algebraically closed field) the characteristic polynomial is a multiple of $\mu_A$; thus $p_A(A)=0$, the Cayley-Hamilton theorem.

The algebraicaly closed field case can be seen to imply the case over arbitrary fields, since neither the characteristic polynomial, not the minimal polynomial, nor the divisibility relation of them changes when the field is extended. If you don't like this flight to algebraicaly closed fields (in fact just a splitting field of the minimal polynomial will suffice), there is an alternative approach that works with cyclic subspaces (spanned by iterated images of a single nonzero vector) instead of eigenspaces. This leads to a basic case involving companion matrices, and an induction argument; see this answer.

Answer 2

For a complex matrix $A,$ the easiest proof I know is to note that the results is true for upper triangular matrices, and then to note that there is a unitary matrix $U$ such that $U^{-1}AU$ is upper triangular. To see the second claim, note that $A$ has an eigenvector $u$ of length $1,$ and complete $\{ u \}$ to an orthonormal basis and proceed by induction. The first claim is a direct computation, as the characteristic polynomial of an upper triangular matrix is visible.

Answer 3

For any diagonal matrix $D$, it is clear that $p_D(D)=0$. Therefore, for any $P \in GL_n(\mathbb{C})$, $p_{PDP^{-1}}(PDP^{-1})=p_D(PDP^{-1})=Pp_D(D)P^{-1}=0$: Cayley theorem holds for diagonalizable matrices. Moreover:

Lemma: the set of diagonalizable matrices is dense in $M_n(\mathbb{C})$.

Proof: Let $M \in M_n(\mathbb{C})$. For some $P \in GL_n(\mathbb{C})$, $M=PT(\lambda_1,\dots,\lambda_n)P^{-1}$ where $T(\lambda_1,\dots,\lambda_n)$ is an upper triangular matrix with diagonal $(\lambda_1,\dots,\lambda_n)$. Let $\epsilon_{ij}$ such that $\lambda_i+\epsilon_{ij} \neq \lambda_k + \epsilon_{kj}$ for $i \neq k$ and $\epsilon_{ij} \underset{j \to + \infty}{\longrightarrow} 0$. Then $PT(\lambda_1+ \epsilon_{1j},\dots,\lambda_n+ \epsilon_{nj})P^{-1}$ is diagonalizable (all their eigenvalues are distincts) and converges to $M$. $\square$

Now $\det$ is continuous, so if $M \in M_n(\mathbb{C})$ and $(M_n)$ is a sequence of diagonalizable matrices converging to $M$, $p_M(M)=\lim\limits_{n \to + \infty} p_{M_n}(M_n)=0$.