# minimal polynomial (endomorphism)

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

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 Cayley-Hamilton theorem, which provides a zero polynomial for $T$.

. 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.

: 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 $degR. 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. 1.

The roots are exactly the eigenvalues of the endomorphism

2. 2.

If the minimal polynomial of $T$ splits into linear factors then $T$ is upper-triangular with respect to some basis

3. 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 $\mathbb{C}$ 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.

Title minimal polynomial (endomorphism) MinimalPolynomialendomorphism 2013-03-22 13:10:14 2013-03-22 13:10:14 mathcam (2727) mathcam (2727) 12 mathcam (2727) Definition msc 15A04 ZeroPolynomial2 OppositePolynomial zero polynomial minimal polynomial