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.

单词	MinimalPolynomialendomorphism
释义	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.
随便看	HollowMatrixRings hollow matrix rings HolmgrenUniquenessTheorem Holmgren uniqueness theorem Holomorphic holomorphic HolomorphicallyConvex holomorphically convex HolomorphicFunctionAssociatedWithContinuousFunction holomorphic function associated with continuous function HolomorphicFunctionsAndLaplacesEquation holomorphic functions and Laplace’s equation HolomorphicFunctionsOfSeveralVariables holomorphic functions of several variables HolomorphicMappingOfCurveAndTangent holomorphic mapping of curve and tangent HolomorphOfAGroup holomorph of a group Homeomorphism homeomorphism homeomorphism Homeomorphism1 HomeomorphismBetweenBooleanSpaces homeomorphism between Boolean spaces HomeomorphismsPreserveConnectedComponents