Generalization of Cayley–Hamilton Theorem for Multivariate Rational Matrices

The two equivalent generalizations of the famous Cayley–Hamilton theorem for multivariate rational matrices over any number field are proposed, which are also generalizations of the nine known generalized Cayley–Hamilton theorems. Restricting our results to a smaller range leads to the two equivalent generalized Cayley–Hamilton theorems for multivariate polynomial matrices over any number field, one of which is more efficient in some cases than a known result.