Semisimple endomorphism

Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be an endomorphism. The map $T$ is semisimple when, for each $T$-invariant subspace $W$ of $V$, there is a $T$-invariant subspace $W'$ with $V = W \oplus W'$.

Remarks

The map $T$ is semisimple if and only if its minimal polynomial has no repeated irreducible factor. Over an algebraically closed field, $T$ is semisimple if and only if it is diagonalisable. The rotation of $\R^2$ by $90$ degrees is semisimple, but it is not diagonalisable over $\R$.

Explanations

L1 human

Semisimple maps generalise diagonalisable maps to fields where eigenvalues can be missing. They have no hidden nilpotent part. Over $\C$, semisimple simply means diagonalisable.