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.