# Algebraic and Geometric Multiplicity

I am having a hard time understand these two concepts

Algebraic multiplicity and Geometric multiplicity of a matrix regarding its eigenvalues

for example

if I have the matrix:

| 5 0 0 |
| 1 5 0 |
| 0 1 5 |


The eigenvalues are 5,5,5, so what does this mean about its multiplicity?

Is geometric multiplicity the number of similar eigenvalue? In this case, 3

and algebraic multiplicity the number of unique eigenvalue? In this case, 1

thanks

• the characteristic equation is det(A-$\lambda$I)=0. And yes if it has n roots then it has algebraic multiplicity of n. – TheBluegrassMathematician Apr 9 '14 at 1:54
• For example, consider the characteristic equation $\lambda^2-4\lambda+4$. Then it does have two roots, but note that it has only have 1 distinct root: $\lambda = 2$. Hence, $\lambda$ would have algebraic multiplicity of 2. – Kaj Hansen Apr 9 '14 at 1:55