[Math] Algebraic and Geometric Multiplicity

Question

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

Answer 1

The geometric multiplicity of an eigenvalue is defined to be the number of linearly independent eigenvectors associated with that eigenvalue.

The algebraic multiplicity of an eigenvalue is defined as the eigenvalue's multiplicity as a root of the characteristic polynomial.