Mariano Suárez-Álvarez

Some time ago, in the process of preparing a talk on the Cayley–Hamilton theorem, I found a slightly silly but somewhat entertaining proof for it involving the theorem on existence of solutions of initial value problems for ordinary differential equations. I wrote it down — the following is in fact the whole text:

A Short Proof of the Cayley–Hamilton Theorem

Let $M$ be an $n\times n$ matrix with real entries and let $\chi_M$ be its characteristic polynomial. The Cayley–Hamilton theorem states that

$\chi_M(M)$ is the zero matrix

and to see this it is enough to show that $\chi_M(M)\cdot x_0=0$ for all $x_0\in\mathbb{R}^n$.

Let us do this. Fix $x_0\in\mathbb{R}^n$. According to the existence theorem for initial value problems for differential equations there is a smooth function $x:U\to\mathbb{R}^n$ defined on some open interval $U\subseteq\mathbb{R}$ containing $0$ such that $x(0)=x_0$ and $x'(t)=M\cdot x(t)$ for all $t\in U$. We can write this last equation in the form $0 = (\partial I-M)\cdot x$, with $I$ the identity matrix and $\partial$ the differentiation operator, and if we multiply by the adjugate matrix $\operatorname{adj}(\partial I-M)$ of $\partial I-M$ we find that \[ 0 = \operatorname{adj}(\partial I-M)\cdot(\partial I-M)\cdot x = \det(\partial I-M)\cdot x = \chi_M(\partial)\cdot x. \] Since $\partial^k\cdot x=M^k\cdot x$ for all $k\in\mathbb{N}_0$, this implies at once that \[ 0 = \chi_M(\partial)\cdot x = \chi_M(M)\cdot x \] and, in particular, if we evaluate the function $\chi_M(M)\cdot x$ at $t=0$ we see that $\chi_M(M)\cdot x_0=0$, which is what we wanted to prove.

I submitted this as filler — those little pieces that show up to literally fill space — to the MAA American Mathematical Monthly and it was rejected on the light of the negative recommendation of the referee. The reviewer's comments on my note were the following:

Reviewer #1: The proof looks like ingenious at a first
sight, but it is cheating. First, the author is subtracting
the differentiation operator (times the identity) and the
matrix M. But these operators act on different spaces, so
this subtraction is not possible (or, at least, should be
explained). Second, and even more important, the key
identity

det(\partial I-M)x=\Xi_M(\partial)x

deserves some explanation. Note that det(\partial I-M) is
not a matrix, but a number (it is the determinant of a
matrix), whereas the factor in the right-hand side,
\Xi_M(\partial), is a matrix. But proving this identity is
like proving the Cayley-Hamilton Theorem! So, in the end,
nothing is proved: the author is hiding the proof of the
Cayley-Hamilton theorem in the previous identity, wrapping
it in a superfluous argument that involves the
differentiation operator. 
  

I find this review amusing, for some value of «amusing». The identity that apparently «hides the proof of the theorem» is nothing more than the definition of the characteristic polynomial, for example. Oh well.