Every square complex matrix is similar to a Jordan form matrix (Five.IV.2.8)

Let $A$ be an $n \times n$ matrix over $\mathbb{C}$. Then $A$ is in Jordan form in the sense of the mission definition: there are a block count $k$, block sizes $sz : \mathrm{Fin}\,k \to \mathbb{N}$, eigenvalues $\lambda : \mathrm{Fin}\,k \to \mathbb{C}$, a reindexing $e$ of the coordinates by the block index type, and an invertible matrix $P$, such that $P^{-1} A P$ is the block-diagonal Jordan matrix built from those blocks, transported along $e$. Equivalently: every square complex matrix is similar to a Jordan form matrix.