To learn something i need a "visualisation" or an "english interpretation" of mathematical ideas, otherwise I'm just memorizing formulas and theorems. And i have a problem with the algebraic multiplicity of Eigenvalues.
As i understand are Eigenvectors vectors that only get stretched by a transformation and don't move their "direction" and they get stretched by associated Eigenwert. The geometric multiplicity is the dimension of the Eigenraum, meaning how many dimensions get only stretched out by the associated Eigenwert.
The algebraic multiplicity is how many times the Eigenwert is a root of the characteristic polynomial. But i don't have a "English" understanding of this fact.
Of course the consequence makes sense: that if the geometric and algebraic multiplicities don't match up a Transformation is not diagonalizable.
The only thing i could think of is to track the number of Eigenvectors needed to create a basis. But it feels like there is more behind the algebraic multiplicity.