Jordan canonical form theorem

Jordan canonical form theorem: Let $T$ be a linear operator on a finite-dimensional vector space $V$ over an algebraically closed field $k$ (more generally, assume the characteristic polynomial splits over $k$). Then there exists a basis of $V$ in which the matrix of $T$ is block diagonal with blocks of the form $J_{m}(\lambda)$, where $\lambda$ ranges over the eigenvalues of $T$. Each Jordan block corresponds to a chain of generalized eigenvectors , and the multiset of block sizes for each $\lambda$ is uniquely determined by $T$ up to permutation.

In this form, $T$ is diagonalizable exactly when all Jordan blocks have size $1$. The sizes of Jordan blocks are governed by the primary decomposition of the $k[x]$-module associated to $T$, and can be derived from the minimal polynomial ; one route is through the rational canonical form theorem

Proof sketch: Decompose $V$ into generalized eigenspaces for each eigenvalue $\lambda$, reducing to the case where $T-\lambda I$ is nilpotent. For a nilpotent operator, build a basis from Jordan chains corresponding to a filtration by kernels of powers; these chains yield Jordan blocks. Combine the decompositions across eigenvalues to obtain the full Jordan form.