Linear mapping/Multiplicities/Introduction/Section

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For a more detailed investigation of eigenspaces, the following concepts are necessary. Let

denote a linear mapping on a finite-dimensional vector space , and . Then the exponent of the linear polynomial inside the characteristic polynomial is called the algebraic multiplicity of , symbolized as . The dimension of the corresponding eigenspace, that is

is called the geometric multiplicity of . Because of fact, the algebraic multiplicity is positive if and only if the geometric multiplicity is positive. In general, these multiplicities might be different, we have however always one estimate.

Let denote a field, and let denote a finite-dimensional vector space. Let

denote a linear mapping and . Then we have the estimate

between the geometric and the algebraic multiplicity.

Let and let be a basis of this eigenspace. We complement this basis with to get a basis of , using fact. With respect to this basis, the describing matrix has the form

The characteristic polynomial equals therefore (using exercise) , so that the algebraic multiplicity is at least .

We consider the -shearing matrix

with . The characteristic polynomial is

so that is the only eigenvalue of . The corresponding eigenspace is


we get that is an eigenvector, and in case , the eigenspace is one-dimensional (in case , we have the identity and the eigenspace is two-dimensional). So in case , the algebraic multiplicity of the eigenvalue equals , and the geometric multiplicity equals .