Let V be a vector spaceMathworldPlanetmath over a field k. Fix a linear transformation T on V. Suppose λ is an eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath of T. The set {vVTv=λv} is called the eigenspaceMathworldPlanetmath (of T) corresponding to λ. Let us write this set Wλ.

Below are some basic properties of eigenspaces.

  1. 1.

    Wλ can be viewed as the kernel of the linear transformation T-λI. As a result, Wλ is a subspacePlanetmathPlanetmathPlanetmath of V.

  2. 2.

    The dimensionPlanetmathPlanetmathPlanetmath of Wλ is called the geometric multiplicity of λ. Let us denote this by gλ. It is easy to see that 1gλ, since the existence of an eigenvalue means the existence of a non-zero eigenvectorMathworldPlanetmathPlanetmathPlanetmath corresponding to the eigenvalue.

  3. 3.

    Wλ is an invariant subspacePlanetmathPlanetmath under T (T-invariant).

  4. 4.

    Wλ1Wλ2=0 iff λ1λ2.

  5. 5.

    In fact, if Wλ is the sum of eigenspaces corresponding to eigenvalues of T other than λ, then WλWλ=0.

From now on, we assume V finite-dimensional.

Let ST be the set of all eigenvalues of T and let W=λSWλ. We have the following properties:

  1. 1.

    If mλ is the algebraic multiplicity of λ, then gλmλ.

  2. 2.

    Suppose the characteristic polynomialMathworldPlanetmathPlanetmath pT(x) of T can be factored into linear terms, then T is diagonalizablePlanetmathPlanetmath iff mλ=gλ for every λST.

  3. 3.

    In other words, if pT(x) splits over k, then T is diagonalizable iff V=W.

For example, let T:22 be given by T(x,y)=(x,x+y). Using the standard basis, T is represented by the matrix


From this matrix, it is easy to see that pT(x)=(x-1)2 is the characteristic polynomial of T and 1 is the only eigenvalue of T with m1=2. Also, it is not hard to see that T(x,y)=(x,y) only when y=0. So W1 is a one-dimensional subspace of 2 generated by (1,0). As a result, T is not diagonalizable.

Title eigenspace
Canonical name Eigenspace
Date of creation 2013-03-22 17:23:07
Last modified on 2013-03-22 17:23:07
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Definition
Classification msc 15A18