minimal polynomial (endomorphism)

Let T be an endomorphismMathworldPlanetmathPlanetmath of an n-dimensional vector spaceMathworldPlanetmath V.

Definitions. We define the , MT(X), to be the unique monic polynomialMathworldPlanetmath of such that MT(T)=0. We say that P(X) is a zero for T if P(T) is the zero endomorphism.

Note that the minimal polynomialPlanetmathPlanetmath exists by virtue of the Cayley-Hamilton theoremMathworldPlanetmath, which provides a zero polynomialMathworldPlanetmath for T.

. Firstly, End(V) is a vector space of dimensionPlanetmathPlanetmath n2. Therefore the n2+1 vectors, iv,T,T2,Tn2, are linearly dependant. So there are coefficientsMathworldPlanetmath, ai not all zero such that i=0n2aiTi=0. We conclude that a non-trivial zero polynomial for T exists. We take MT(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 MT(X)P(X).


By the division algorithmPlanetmathPlanetmath for polynomialsMathworldPlanetmathPlanetmath, P(X)=Q(X)MT(X)+R(X) with degR<degMT. We note that R(X) is also a zero polynomial for T and by minimality of MT(X), must be just 0. Thus we have shown MT(X)P(X). ∎

The minimal polynomial has a number of interesting properties:

  1. 1.

    The roots are exactly the eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath 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 algebraMathworldPlanetmath that every endomorphism of a finite dimensional vector space over may be upper-triangularized.

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

In general they may not be the same - for example any diagonal matrixMathworldPlanetmath 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