# Thread: Algebraic and Geometric multiplicity

1. ## Algebraic and Geometric multiplicity

let A be the matrix
|4 1 0 0 0|
|0 4 1 0 0|
|0 0 4 1 0|
|0 0 0 4 1|
|0 0 0 0 4|

I am asked to find the algebraic multiplicity
and the geometric multiplicity

I think as it is an upper triangular matrix eigenvalues are the diagonal entries i.e. all 4,
4^5 so Am is 5, im not sure on the geometric multiplicity though. Any help appreciated!

2. The geometric multiplicity is the dimension of the eigenspace corresponding to your eigenvalue. That is, when you solve

$(A-4 I)x=0,$

how many arbitrary parameters do you get? That'll be the geometric multiplicity.

3. Ah ok, I've done that and I find x2=x3=x4=x5=0 leaving only arbitrary parameter x1 hence Gm is 1 and eigen-space is Sp(e1). Is that right?

4. Sounds good to me. You can check your work here.