The algebraic multiplicity of λi is the degree of the root λi in the polynomial det(A−λ).

The geometric multiplicity is the dimension of the eigenspace of eigenvalue λi.For example:

[1101] has root 1 with algebraic multiplicity 2, but the geometric multiplicity 1.My question is : why geometric multiplicity is always bounded by algebraic multiplicity?

Thanks.

**Answer**

Suppose the geometric multiplicity of the eigenvalue λ of A is k. Then we have k linearly independent vectors v1,…,vk such that Avi=λvi. If we change our basis so that the first k elements of the basis are v1,…,vk, then with respect to this basis we have

A=(λIkB0C)

where Ik is the k×k identity matrix. Since the characteristic polynomial is independent of choice of basis, we have

charA(x)=charλIk(x)charC(x)=(x−λ)kcharC(x)

so the algebraic multiplicity of λ is at least k.

**Attribution***Source : Link , Question Author : Jack2019 , Answer Author : Alex Becker*