How do you calculate the geometric multiplicity

of eigenvectors and eigenvalues, in my course it says: " it is the dimension of eigenspace" but I have no idea how I can calculate that...

You should know that there is a difference between algebraic multiplicity and geometric multiplicity.

If the characteristic polynomial for some matrix is for example

p(饾泴)=(饾泴-3)^2(饾泴-1)^3
The algebraic multiplicity of the eigenvalues is 2 for 饾泴=3 and 3 for 饾泴=1.

To find the eigenvectors you solve the matrix equation

Ax=饾泴x for each 饾泴.
The geometric multiplicity is the number of linearly independent eigenvector associated with each 饾泴 after solving the above matrix equation.

In other words, the geometric multiplicity of an eigenvalue 饾泴 of a matrix A is the dimension of the subspace of vectors x for which Ax = 饾泴x.This vector subspaces is called the eigenspace E饾泴.
The algebraic multiplicity of an eigenvalue鈮?geometric multiplicity.

Suppose we have a 2x2 matrix A.

If we are to work out the eigenvalues of this matrix we would find the determinant of A and hence deduce a quadratic equation. Quadratic equations generally lead to two roots, one repeated root or a complex conjugate pair of roots.
Suppose our matrix yielded the following eigenvalues 位1 = 2 and 位2 = 3
Then we can just say matrix A has eigenvalues 2 (with multiplicity 1) and 3 (again with multiplicity 1)

However suppose the matrix gave us 位1 = 1 and 位1 = 1 for example.
Then we could just say matrix A has an eigenvalue 1 with multiplicity 2.
In a way multiplicity can be seen as how many times the solution appears.

I hope this helps.