Let $F$ be a field, let $V$ be a nonzero finite-dimensional [vector space](/page/Vector%20Space) over $F$, and let $T: V \to V$ be a linear operator. Suppose that the [characteristic polynomial](/page/Characteristic%20Polynomial) of $T$ splits over $F$ as
where $r \in \mathbb{N}$, where $\lambda_1,\ldots,\lambda_r \in F$ are distinct, and where $m_1,\ldots,m_r \in \mathbb{N}$. For each $j \in \{1,\ldots,r\}$, let