minimal polynomial

To understand the influence of repeated factors in the characteristic polynomial, it is worth considering whether their presence is essential for generating a zero matrix, in the sense of whether the product of the remaining factors is already zero, or it is not and the additional factor is required to realize the Cayley-Hamilton theorem. In other words, it could happen that there were other vanishing polynomials for the matrix $M$, say $\phi(M) = O$ and $\psi(M) = O$. They could always be normalized to become monic. But, by long division, one of them, depending on their relative degrees, would be a multiple of the other with a remainder:

$$\begin{eqnarray*} \phi(M) & = & \psi(M) \sigma(M) + \rho(M). \end{eqnarray*}$$

Since both $\phi(M)$ and $\psi(M)$ vanish, so must $\rho(M)$. The upshot of this is that there will always be a unique monic polynomial of least degree, $\mu(M)$, satisfied by any given matrix. Accordingly it would be called the minimal polynomial of the matrix, and ought to be used in place of the characteristic polynomial when discussing eigenmatrices.

Even so, the minimal polynomial may still have repeated roots, which means that there is a nontrivial chain of matrices, none of them zero, which map from one to another in sequence, and finally to zero. Their rows and columns must have the same property, which could be exploited in forming a basis and establishing a canonical form for the matrix.