Minimal polynomial/Diagonal matrix/Different entries/Example

For a diagonal matrix

${}M={\begin{pmatrix}d_{1}&0&\cdots &0\\0&d_{2}&\ddots &\vdots \\\vdots &\ddots &\ddots &0\\0&\cdots &0&d_{n}\end{pmatrix}}\,$

with different entries ${}d_{i}$, the minimal polynomial is

${}P=(X-d_{1})(X-d_{2})\cdots (X-d_{n})\,.$

This polynomial is sent under the substitution to

$(M-d_{1}E_{n})\circ (M-d_{2}E_{n})\circ (M-d_{n}E_{n}).$

We apply this to a standard vector ${}e_{i}$. Then factor ${}(M-d_{j}E_{n})$ sends ${}e_{i}$to ${}(d_{i}-d_{j})e_{i}$. Therefore, the ${}i$-th factor ensures that ${}e_{i}$ is annihilated. Since a basis is mapped under ${}P(M)$ to ${}0$, it must be the zero mapping.

Assume now that ${}P$ is not the minimal polynomial ${}\mu$. Then there exists, due to fact, a polynomial ${}Q$ with

${}P=Q\mu \,,$

and, because of fact, ${}\mu$ is a partial product of the linear factors of ${}P$. But as soon as one factor of ${}P$ is removed, say we remove ${}X-d_{i}$, then ${}e_{i}$ is not annihilated by the corresponding mapping.