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We introduce the notion of characteristic function of a quaternionic matrix,whose roots are the left eigenvalues. We prove that for all $2\times 2$matrices and for $3\times 3$ matrices having some zero entry outside thediagonal there is a characteristic function which is a polynomial. For theother $3\times 3$ matrices the characteristic function is a rational functionwith one point of discontinuity. We prove that Hamilton-Cayley theorem holds inall cases.

Characteristic functions and Hamilton-Cayley theorem for left eigenvalues of quaternionic matrices