Let $T$ be an endomorphism of an $n$ dimensional vector space $V$

Definitions. We define the minimal polynomial, $M_T(X)$ to be the unique monic polynomial of minimal degree such that $M_T(T) = 0$ We say that $P(X)$ is a zero polynomial for $T$ if $P(T)$ is the zero endomorphism.

Note that the minimal polynomial exists by virtue of the Cayley-Hamilton theorem, which provides a zero polynomial for $T$

Properties. Firstly, $\operatorname{End}(V)$ is a vector space of dimension $n^2$ Therefore the $n^2 + 1$ vectors, $i_v, T, T^2, \ldots T^{n^2}$ are linearly dependant. So there are coefficients, $a_i$ not all zero such that $\sum_{i=0}^{n^2} a_i T^i = 0$ We conclude that a non-trivial zero polynomial for $T$ exists. We take $M_T(X)$ to be a zero polynomial for $T$ of minimal degree with leading coefficient one.

Lemma: If $P(X)$ is a zero polynomial for $T$ then $M_T(X) \mid P(X)$

Proof. By the division algorithm for polynomials, $P(X) = Q(X)M_T(X) + R(X)$ with $deg R < deg M_T$ We note that $R(X)$ is also a zero polynomial for $T$ and by minimality of $M_T(X)$ must be just $0$ Thus we have shown $M_T(X) \mid P(X)$

The minimal polynomial has a number of interesting properties:

1. The roots are exactly the eigenvalues of the endomorphism
2. If the minimal polynomial of $T$ splits into linear factors then $T$ is upper-triangular with respect to some basis
3. The minimal polynomial of $T$ splits into distinct linear factors (i.e. no repeated roots) if and only if $T$ is diagonal with respect to some basis.

It is then a simple corollary of the fundamental theorem of algebra that every endomorphism of a finite dimensional vector space over $\mathbb{C}$ may be upper-triangularized.

The minimal polynomial is intimately related to the characteristic polynomial for $T$ For let $\chi_T(X)$ be the characteristc polynomial. Since $\chi_T(T)=0$ we have by the above lemma that $M_T(X) \mid \chi_T(X)$ It is also a fact that the eigenvalues of $T$ are exactly the roots of $\chi_T$ So when split into linear factors the only difference between $M_T(X)$ and $\chi_T(X)$ is the algebraic multiplicity of the roots.

In general they may not be the same - for example any diagonal matrix with repeated eigenvalues.