Let $V$ be a finite-dimensional vector space over an algebraically closed field $k$, and $T: V \to V$ a linear operator. By the structure theorem for $k[x]$-modules:

$V \cong k[x]/(x - \lambda_1)^{e_1} \oplus \cdots \oplus k[x]/(x - \lambda_s)^{e_s},$

where the $\lambda_i$ are eigenvalues of $T$ (not necessarily distinct) and $e_i \geq 1$. Each summand $k[x]/(x-\lambda)^e$ corresponds to a Jordan block:

$J_e(\lambda) = \begin{pmatrix} \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \cdots & 0 \\ \vdots & & \ddots & \ddots & \vdots \\ 0 & & & \lambda & 1 \\ 0 & & & & \lambda \end{pmatrix} \in k^{e \times e}.$

The Jordan form $J = \mathrm{diag}(J_{e_1}(\lambda_1), \ldots, J_{e_s}(\lambda_s))$ is unique up to permutation of blocks.

Over any field $k$ (not necessarily algebraically closed), the invariant factor decomposition gives:

$V \cong k[x]/(f_1) \oplus \cdots \oplus k[x]/(f_t),$

with $f_1 \mid f_2 \mid \cdots \mid f_t$ monic polynomials. Each $k[x]/(f_i)$ has a basis in which $T$ acts as the companion matrix of $f_i$. The last invariant factor $f_t$ is the minimal polynomial, and $\prod f_i$ is the characteristic polynomial.