In this paper, the problem of simultaneous diagonalization of m-tuples of n-order square complex matrices, is analyzed and some necessary and some necessary and sufficient conditions for this property to be fulfilled are presented. This study has an interest in its applications in different areas as for example in engineering and physical sciences. For example, they appear founding when we must give the instanton solution of Yang-Mills field presented in an octonion form, and it can be represented by triples of traceless matrices. In the case where the m-tuple does not simultaneously diagonalize, the possibility of to find near of the given m-tuple, an m-tuple that diagonalize simultaneously is studied

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