Eigenvalue (synthesis)

The concepts of clean Vector, eigenvalue, and clean space apply to endomorphisms, i.e. linear applications of a vector Space in itself. They are closely bound, and form a pillar of the reduction of the endomorphisms, left the Linear algebra which aims at all in all breaking up in the most effective possible way space direct of stable subspaces.

This article summarizes the essential mathematical properties associated with these concepts and returns towards articles dedicated for a deepening.

the article Eigenvalue, clean vector and clean space as well details the history of these concepts like their uses in the field of the Mathématiques as in the other scientific branches .

Definitions and properties

Subsequently, one considers a vector space E on a commutative body \ mathbb K. In practice, the body is generally the body of the complexes \ mathbb C and the vector space is of finished size. One will specify in each section, the possible restrictions on the body or dimension. One will note U a endomorphism of E and Id the endomorphism identity.

Eigenvalue

Either \ lambda a scalar of \ mathbb K, \ lambda is a eigenvalue of the endomorphism U if and only if there exists a vector X not no one such as u (X) = \ lambda X. The whole of the eigenvalues of U is thus the whole of the scalars \ lambda such as u- \ lambda Id is not injective (in other words its core is not tiny room to the null vector).

If E is of dimension N , then U has with more N eigenvalues.

Examples:

  • if U = Id then U has only one eigenvalue: 1.
  • if U is defined on \ R^2 by u (x_1, x_2) = (x_1 + 6x_2; x_1 + 2x_2) \, then U has two eigenvalues
    • 4 because U (2; 1) = (8; 4) = 4 (2; 1)
    • -1 because U (- 3; 1) = (3; -1) = -1 (- 3; 1)
    • not of another eigenvalue since dimension is 2.
One also speaks about eigenvalue of a square matrix. Indeed, a square matrix is the representation of a endomorphism on \ mathbb K^n, relative at the canonical base.

Clean vector

Either X a vector not no one of E , X is a clean vector of the endomorphism U if and only if there exists a scalar \ lambda such as u (X) = \ lambda x. It will be said that X is a clean vector associated with the eigenvalue \ lambda.
  • a clean vector cannot be associated with two eigenvalues different
  • a family from K clean vectors associated with K different eigenvalues constitutes a free Famille.
In the case of an eigenvalue associated with a square matrix, one often employs the term of clean column rather than that of clean vector.

Clean subspaces

That is to say λ an eigenvalue of U ; then the unit made up of the clean vectors of eigenvalue λ , and of the null vector, forms a vectorial subspace of E called clean subspace of U associated with the eigenvalue λ .
  • Is λ an eigenvalue, then the clean space of eigenvalue λ is the vectorial subspace equal to the core of U - λ.Id .
  • By definition of an eigenvalue, a clean space is never tiny room to the null vector.
  • clean spaces Ei of eigenvalues λi form a direct Somme of stable vectorial subspaces by U .
This property is shown simply with the tools developed in the article Polynôme of endomorphism. X - λi is a polynomial canceler of the restriction of U on Ei . These polynomials are very first between them, the last proposal of the paragraph Idéaux cancelers finishes the demonstration.
the direct sum does not make therefore the Ei of the additional subspaces. It will be the case only when the endomorphism is diagonalisable.
  • If two endomorphisms U and v commutate, then any clean space of U is stable by v .
Is X a clean vector of U , of eigenvalue λ ; then:
U \ circ v (X) \; = \; v \ circ U (X) \; = \; \ lambda v (X)
We thus showed that v (X) is either null or clean vector of eigenvalue λ : it is thus well element of the clean space of λ .

Characteristic polynomial

See also: characteristic Polynomial

One places here within the framework of a vector space E of finished size N .

One calls polynomial characteristic of the endomorphism U , the formal polynomial \ det (U - X.Id)

  • the roots of the characteristic polynomial are the eigenvalues of U .
One of the properties of the determinant is to be null if and only if the core of the associated endomorphism is nonreduced to the null vector. That means that λ is root of the polynomial characteristic if and only if the core of U - λ.Id is nonreduced to the null vector, which is a requirement and sufficient so that λ is an eigenvalue.
  • Is has a Isomorphisme, i.e. a bijective endomorphism , then if U is a endomorphism, U and a-1.u.a has even characteristic polynomial and thus same eigenvalues.
Indeed, the article on the determining shows us that:
\ det (a^ {- 1} (U \ lambda I) a) \; = \; \ det (a^ {- 1} ua- \ lambda a^ {- 1} Ia) \; = \; \ det (a^ {- 1} ua- \ lambda I)
  • If \ mathbb K is algebraically closed or if it is equal to the body of the real numbers and that N is odd, then U has at least an eigenvalue.
To say that a body is algebraically closed, it is to say that any polynomial of degree different from zero admits at least a root. This root is necessarily an eigenvalue, according to the first property of this paragraph. Moreover, the real polynomials always have a root if the degree is odd. That finishes the demonstration. The algebraic order of multiplicity of an eigenvalue is the order of multiplicity of the root in the characteristic polynomial. The algebraic order of multiplicity of an eigenvalue λ thus corresponds to the power of the students' rag procession (X-λ) in the characteristic polynomial.
  • In a body algebraically closed,
    • the determinant is equal to the product of the high eigenvalues to their order of algebraic multiplicity.
    • the trace is equal to the sum of the eigenvalues multiplied by their order of algebraic multiplicity.

Minimal polynomial

See also: minimal Polynomial

One places oneself here in finished dimension. The minimal polynomial is the standardized polynomial (its factor moreover high degree is equal to 1) of smaller degree which cancels the endomorphism U

  • the Théorème of Cayley-Hamilton makes it possible to affirm than the minimal polynomial divides the characteristic polynomial
  • the roots of the minimal polynomial form the whole of the eigenvalues of U

Characteristic subspaces

See also: characteristic Subspace

One places in a vector space E of size finished N on a body \ mathbb K algebraically closed.

If \ lambda_i is an eigenvalue of U , whose order of multiplicity is \ alpha_i, one calls subspace characteristic of U associated with the eigenvalue \ lambda_i the core with (U - \ lambda_i Id) ^ {\ alpha_i}. One will note the characteristic subspace E_i

  • E_i is also the core of (U - \ lambda_i Id) ^ {\ beta_i} where \ beta_i is the order of multiplicity de \ lambda_i in the minimal polynomial.
  • E_i is stable by U
  • \ dim (E_i) = \ alpha_i
  • space E is direct sum of its subspaces characteristic
  • the restriction of U on E_i has as a minimal polynomial (X \ lambda_i) ^ {\ beta_i}

Reduction of endormorphism

See also: Reduction of endomorphism

One will place oneself in finished dimension. The study of the eigenvalues makes it possible to find a form simpler of the endomorphisms, it is what is called their reduction.

Diagonalisation

See also: Diagonalisation, Matrix diagonalisable

There exists a particular case where the knowledge of the clean vectors and eigenvalues associated armature the exhaustive behavior with the endomorphism. It is the case when the endomorphism is diagonalisable, i.e. there exists a base of clean vectors. Numerical examples are given in the article Matrice diagonalisable. The following criteria all are of the requirements and sufficient so that a endomorphism is diagonalisable:

  • There exists a base of clean vectors

  • the sum of clean spaces generates whole space
  • the sum of dimensions of clean spaces is equal to the dimension of whole space
  • the minimal polynomial is divided on K and with simple roots. ( Demonstration in Polynomial of endomorphism. )
  • any space clean has a dimension equal to the algebraic multiplicity of the associated eigenvalue.
  • any representation matric M of U is diagonalisable, i.e. can be respectively written in the form M=P^ {- 1} DP with P and D matrices invertible and diagonal.

To these equivalent properties are added the following implications:

  • If there exists N distinct eigenvalues, then the endomorphism is diagonalisable.
  • If the endomorphism is diagonalisable, then the characteristic polynomial is divided.

In the case (i.e. where the body of number is that of the complexes) this property complexes is Presque everywhere true within the meaning of measurement. Within the meaning of the Topologie the whole of the endomorphisms diagonalisables is dense.

Decomposition of Dunford

See also: Decomposition of Dunford

  • Is U a endomorphism of E. If U admits a divided minimal polynomial, then it can be written in the form U = d+n with D diagonalisable and N nilpotent such as d.n=n.d. Moreover D and N are polynomials out of U.

Representation of Jordan

See also: Reduction of Jordan

One places oneself in a vector space on \ mathbb K algebraically closed.

The representation of Jordan proves that any endomorphism U on E is trigonalisable. It shows that any reduction of the endomorphism to characteristic space E_i associated with the eigenvalue \ lambda_i has a formed representation of blocks of the following form

\ mathcal {J} _ {\ lambda} = \ begin {bmatrix} \ lambda_i & 1 & & & & \ \ & \ lambda_i & 1 & & (0) & \ \ & & \ ddots & \ ddots & & \ \ & & & \ ddots & \ ddots & \ \ & (0) & & & \ lambda_i & 1 \ \ & & & & & \ lambda_i \ \ \end{bmatrix} called block of Jordan and that the endomorphism has a matric representation in the form
\begin{bmatrix} \ mathcal {J} _ {\ lambda_1} & & & & \ \ & \ mathcal {J} _ {\ lambda_2} & & & \ \ & & \ ddots & & \ \ & & & \ ddots & \ \ & & & & \ mathcal {J} _ {\ lambda_r} \ \ \end{bmatrix} where the scalars \ lambda_i are the eigenvalues of the endomorphism considered and \ mathcal {J} _ {\ lambda_i} a block of associated Jordan.

See too

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