PlanetMath: minimal polynomial (endomorphism)

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minimal polynomial (endomorphism)

(Definition)

Let be an endomorphism of an -dimensional vector space .

Definitions. We define the minimal polynomial, , to be the unique monic polynomial of minimal degree such that . We say that is a zero polynomial for if is the zero endomorphism.

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

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

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

Proof. By the division algorithm for polynomials,

with

. We note that

is also a zero polynomial for

and by minimality of

, must be just 0. Thus we have shown $M_T(X) \mid P(X)$ . $\qedsymbol$

The minimal polynomial has a number of interesting properties:

The roots are exactly the eigenvalues of the endomorphism
If the minimal polynomial of splits into linear factors then is upper-triangular with respect to some basis
The minimal polynomial of splits into distinct linear factors (i.e. no repeated roots) if and only if 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 . 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 are exactly the roots of $\chi_T$ . So when split into linear factors the only difference between and $\chi_T(X)$ is the algebraic multiplicity of the roots.