 Solution

The primary decomposition theorem for linear maps starts with a linear map $T:V\to V$ on a finite-dimensional vector space over a field $F$. Factor its minimal polynomial aswhere the $p_j$ are distinct monic polynomials that are irreducible polynomials and $a_j\ge 1$. Define $V_j=\ker p_j(T{)}^{a_j}$. ThenThus the direct sum separates the irreducible polynomial factors into uniquely determined invariant subspaces. The statement works even when the minimal polynomial does not split into linear polynomials over $F$.