minimal polynomial (endomorphism)
Let T be an endomorphism of an n-dimensional vector space V.
Definitions. We define the , MT(X), to be the unique monic polynomial 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 polynomial exists by virtue of the Cayley-Hamilton theorem, which provides a zero polynomial for T.
. Firstly, End(V) is a vector space of dimension n2. Therefore the n2+1 vectors, iv,T,T2,…Tn2, are linearly dependant. So there are coefficients, ai not all zero such that ∑n2i=0aiTi=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).
Proof.
By the division algorithm for polynomials, 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.
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 ℂ may be upper-triangularized.
The minimal polynomial is intimately related to the characteristic polynomial 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 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 |