What are the relations between geometric multiplicity and algebraic multiplicity, concerning diagonalizability?

Question

I know that gemu <= almu and that if the sum of the gemu of all eigenvalues is equivalent to the dimension of the vector space, then the linear transformation is diagonalizable, but is there any direct connections between gemu, almu, and diagonalizability?

Answer 1

For an $n\times n$ matrix $M$ to be diagonalizable, we need $n$ linearly independent eigen vectors.

If each eigen value gives us as many eigenvectors as its algebraic multiplicity, we are done, otherwise we are not able to diagonalize our matrix.