Jordan canonical form theorem

Let $T$ be a linear operator on a finite-dimensional vector space $V$ over a field $k$ . If the characteristic polynomial of $T$ splits over $k$ , then $V$ has a basis in which the matrix of $T$ is block diagonal with Jordan blocks $J_m(\lambda)$ . The eigenvalues $\lambda$ and, for each $\lambda$ , the multiset of block sizes are uniquely determined by $T$ , up to permuting the blocks.

Consequences

Each block corresponds to a chain of generalized eigenvectors. The operator $T$ is diagonalizable exactly when every block has size $1$ , and the largest block size for $\lambda$ is the exponent of $x-\lambda$ in the minimal polynomial.