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 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 $degR<degM_{T}$ . 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 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)
Canonical name	MinimalPolynomialendomorphism
Date of creation	2013-03-22 13:10:14
Last modified on	2013-03-22 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