Jordan canonical form theorem

A Jordan block or Jordan matrix is a matrix of the form

with a constant value $\lambda$ along the diagonal and 1’s on the superdiagonal. Some texts the 1’s on the subdiagonal instead.

Theorem.

Let $V$ be a finite-dimensional vector space over a field $F$ and $t\mathrm{:}V\mathrm{\to }V$ be a linear transformation. Then, if the characteristic polynomial factors completely over $F$, there will exist a basis of $V$ with respect to which the matrix of $t$ is of the form

where each $J_i$ is a Jordan block in which $\lambda \mathrm{=}{\lambda }_i$.

The matrix in Theorem 1 is called a Jordan canonical form for the transformation t.