# Interpretation of algebraic multiplicity

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.

Perhaps a simple example would help. Suppose you have a plane with origin and coordinate system. The mapping that projects all points onto the $x$-axis is linear. It was eigenvalues $0$ and $1$. The $y$-axis goes with eigenvalue $0$ and the $x$-axis goes with eigenvalue $1$. Now suppose after the projection we rotate by a right angle so all points end up on the $y$-axis. If we do this rotated projection twice we map all points to the origin. This means that the rotated projection has an eigenvalue of $0$ with algebraic multiplicity $2$. However, only the $y$-axis is mapped to the origin and therefore the geometric multiplicity of the eigenvalue $0$ is $1$. That is, all points not on the $y$-axis get mapped to non origin points on the $y$-axis and hence can't be eigenvectors because the direction is changed.