When is a matrix diagonalizable?

Forward direction. If $A = PDP^{-1}$ with $D$ diagonal, then the columns of $P$ are eigenvectors of $A$ (since $AP = PD$). They form a basis of $\mathbb{R}^n$, so the sum of the dimensions of the eigenspaces is at least $n$. But each eigenspace is a subspace, and their sum is direct (eigenspaces for distinct eigenvalues are in direct sum). So the sum of dimensions is exactly $n$.

Converse. Conversely, if the sum of the dimensions of the eigenspaces equals $n$, choose a basis of each eigenspace. The union of these bases has $n$ vectors, and since eigenspaces for distinct eigenvalues are in direct sum, the union is a basis of $\mathbb{R}^n$. In this basis, the matrix of $A$ is diagonal. So $A$ is diagonalizable.

Remark. This criterion is equivalent to saying that the characteristic polynomial of $A$ splits and the algebraic multiplicity of each eigenvalue equals its geometric multiplicity.