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In this paper we extend the notion of a locally hypercyclic operator to thatof a locally hypercyclic tuple of operators. We then show that the class ofhypercyclic tuples of operators forms a proper subclass to that of locallyhypercyclic tuples of operators. What is rather remarkable is that in everyfinite dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$, a pair ofcommuting matrices exists which forms a locally hypercyclic, non-hypercyclictuple. This comes in direct contrast to the case of hypercyclic tuples wherethe minimal number of matrices required for hypercyclicity is related to thedimension of the vector space. In this direction we prove that the minimalnumber of diagonal matrices required to form a hypercyclic tuple on$\mathbb{R}^n$ is $n+1$, thus complementing a recent result due to Feldman.

On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple