A symmetrization of the Jordan canonical form

For a (finite or infinite dimensional) vector space V, the notion of a symmetric Jordan canonical form of an operator T∈L(V) having a minimal polynomial is defined and used to verify the relation between the notions of “Jordan canonical form” and “rational canonical form.” The paper extends and repairs Theorem 2.2 of Radjabalipour (2013) [6]. In particular, it is shown that there exists an auxiliary nilpotent operator S∈L(W), depending on T, such that every Jordan canonical form of S yields a symmetric Jordan canonical form and, if the characteristic of the ground field is zero, a rational canonical form for T. The paper concludes with a direct proof of the symmetric Jordan canonical form which “integrates” into a rational canonical form.