Unitary Diagonalization and Quadratic Forms

Abstract

As we saw in Chap. 8, when V is a finite-dimensional vector space over ${\mathbb {F}}$, then a linear mapping $T:V\rightarrow V$ is semisimple if and only if its eigenvalues lie in ${\mathbb {F}}$ and its minimal polynomial has only simple roots. It would be useful to have a result that would allow one to predict that T is semisimple on the basis of a criterion that is simpler than finding the minimal polynomial, which, after all, requires knowing the roots of the characteristic polynomial.