minimal polynomial (endomorphism)
Let $T$ be an endomorphism^{} of an $n$dimensional vector space^{} $V$.
Definitions. We define the , ${M}_{T}(X)$, to be the unique monic polynomial^{} of such that ${M}_{T}(T)=0$. We say that $P(X)$ is a zero for $T$ if $P(T)$ is the zero endomorphism.
Note that the minimal polynomial^{} exists by virtue of the CayleyHamilton theorem^{}, which provides a zero polynomial^{} for $T$.
. Firstly, $\mathrm{End}(V)$ is a vector space of dimension^{} ${n}^{2}$. Therefore the ${n}^{2}+1$ vectors, ${i}_{v},T,{T}^{2},\mathrm{\dots}{T}^{{n}^{2}}$, 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 nontrivial zero polynomial for $T$ exists. We take ${M}_{T}(X)$ to be a zero polynomial for $T$ of minimal degree with leading coefficient one.
: 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 $$. 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 uppertriangular 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 corollary of the fundamental theorem of algebra^{} that every endomorphism of a finite dimensional vector space over $\u2102$ may be uppertriangularized.
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.
Title  minimal polynomial (endomorphism) 

Canonical name  MinimalPolynomialendomorphism 
Date of creation  20130322 13:10:14 
Last modified on  20130322 13:10:14 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  12 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 15A04 
Related topic  ZeroPolynomial2 
Related topic  OppositePolynomial 
Defines  zero polynomial 
Defines  minimal polynomial 