Algebraic and Geometric Multiplicity

Algebraic multipicity is number of identical roots of characteristic equation. Geometric multiplicity is the number of independent eigenvectors corresponding to any multiple root. If algebraic multiplicity equals geometric multiplicity, A can be diagonalized.

In the OP 0 is a root of algebraic multiplicity 2 and 1 is a root of algebraic multiplicity 2

If algebraic multiplicity is k, geometric multiplicity is k if

Rank | A-Iλ_{k} | is n-k (Mirsky, pg 204),

or just try to find k independent eigenvectors corresponding to λ_{k}. If, for example, k=2 and you can only find one independent eigenvector, geometric multiplicity is less than algebraic multiplicity and you can’t diagonalize A.